From the description of Charisma in the Basic Rules:

Charisma measures your ability to interact effectively with others. It includes such factors as confidence and eloquence, and it can represent a charming or commanding personality.

You can say the sweetest laced words in the world, but if you say them condescendingly or without the presence to gain people's attention it wont amount to much.

As a DM, I would have him define the mannerisms of his character to reflect why he has a 9 charisma. Does he say smooth things but mumbles them so no one can hear them? (Gordon Agrippa from Black Clover is an excellent example of this type of character)

Or does he say them in condescending ways? Much like how a high elf would talk to/about a dwarf. Characters with a 9 charisma have some sort of personality flaw that slightly detracts from their interpersonal skills. You can try having your DM implement a rule that you can dictate what your character says, but your Charisma roll reflects HOW he says it.

You can call a King a fool with a charisma roll of 20 and he will think you a jester and laugh, but just try calling a King a terrific ruler with a charisma roll of 1 - he will think you are trying to insult him in some way and have you arrested!

Resolve it the same way for all player/character disparity. What's written on the character sheet takes precedence

Players don't have to be strong or dexterous to play characters with high STR and DEX. They don't have to know how to skin an animal or track game to play a character with Survival proficiency. Most obviously - they don't have to be able to do magic to play a spellcaster!

To determine success in a challenge, the numbers on the character sheet have higher precedence than player skill.

If a player who is really bad at tactics is playing a character with a lot of skill in that area, the GM should take the horrible plans the player comes up with and assume that the character has a good plan.

It works the same the other way, though this requires a fair amount of trust at the table. If a player comes up with a great battle plan but their character is unskilled then the GM should take the great plan and have it fail in some way, to reflect lack of character skill. This has to be done delicately, and consistently, to avoid hurting feelings.

Another option is that when one player has a cleaver idea that their character probably wouldn't come up with, let the characters of other players use it.

My experience

I've been in two games with exactly this situation.

In one case, the player wasn't aware they were doing this and a quick OOC conversation sorted the issue. "Hey, Bob, I know you are a smooth talker but your character isn't, so please don't get upset when your character flubs conversations."

In the second case, the player was doing it deliberately and had done so in previous games (board, card, and rpg). The conversation at the table got heated (as if often does when you call out a manipulator) and ended in the player leaving. It was upsetting at the time, but in hindsight it was the right thing to do.

The problem issue also happens with very smart players playing low-Int characters but solving the puzzle in seconds, while the 18-Int Wizard's average-IQ player sits there still trying to understand what the setup was. It's true of all the "mental"/"personality" stats.

One solution I have seen that skips around a bunch of discussion and rolls (like where the adroit player's PC tries something they're poor at with help from the PC that probably should have done it) is for the DM to allow success without a roll on a good attempt, but to narrate it thusly:

"The guard is unmoved by your explanation, as your voice squeaks and you stumble over a word, making you sound suspicious, but your companion (DM indicates high-Charisma character) sees what you're trying to do and explains it more convincingly."

[If the second character has not stated they're at least helping, then this sort of DMing may require some level of agreement from players that it was okay to do it (e.g. agreeing its fine if it wasn't otherwise consequential, in order to keep the game moving ahead while making sense).]

The problem isn't the DM.

Inverse My Guy Syndrome is as much of a challenge to overcome as My Guy Syndrome. The tone of your question was (you to the other player) "your guy wouldn't do that" or "your guy couldn't do that."

Actually, yes they can and yes they could. It's the DM's place to set the boundaries.

Stats are descriptive, not prescriptive.

Unlike editions of the game where not having skill points might block you from attempting something, any player can try anything, and the DM narrates the results. Dice are only rolled when necessary. (Basic Rules, p. 4).

1. The DM describes the environment.




2. The players describe what they want to do.




3. The DM narrates the results of the adventurers’ actions. Describing the results often leads to another decision point, which brings the flow of the game right back to step 1.

It is OK for a player to have skill

It is counterproductive to presume that as soon as the game starts the player somehow disappears and only the character is present - that immersion into a given character in an RPG happens to the extent that any player desires it to. The level of immersion into "in-character", both preferred and experienced, varies with each player and with each table. The characters in an RPG only exist and have agency because a player is driving them for the game session(s).

• As an example of how you can't avoid a bit of meta-play: the game feat Lucky, and the Wizard (Divination School) ability Portent, are Meta Game Constructs. They give die roll control to a player/character. The dice themselves are a meta construct outside of the characters who are inside-the-fictional setting of the game.

A related meta problem is this²:

... mental stats are problematic in role-playing games because the
whole point of a role-playing game is for the player to make choices
and solve problems.

I have added an at-table example of player/character juxtaposition, as regards low versus high Charisma, that troublesome stat, at the end of this answer.

Don't resent other player's ability - keep working on your ability

Mike Mornard once made the cogent observation - referring to the game Diplomacy's influence on early D&D - that player skill is a part of the game. It still is. The presence of min-max and optimizer play styles attests to that. So does the learning curve of new players as they become experienced players. Player skill is a part of the game, even if it is seen by some a being part of the meta-game.

• When someone makes a spell choice on level up, what informs the
  reason for that choice: character growth or optimization? Are you
  going to complain about that choice to the DM, or tell the player
  that choosing that spell isn't what their character can, or would,
  do? I doubt it.

If the DM is happy to adjudicate the social encounter based on a clever bit of wordplay, good! That's fun and (1) it is good role play, (2) players play the game to have fun, and (3) in this edition you can try anything and see if you succeed.¹ That third point is a fundamental premise of the 5e ability / skill set up. Beyond that, stats don't rule all; see advantage and disadvantage rules as an example.

• 
  

  Advantage beats stats; in a case like this it can (depending on the circumstances) overwrite stats: if a given verbal exchange is particularly good or clever, it is a perfect place for the DM to apply advantage (+5 boost, roughly) to any attempt) due to a circumstance. Note: circumstances are not dictated by ability score and stats and this won't happen all of the time. To earn, for a given encounter, +5 is equivalent to the difference between an 18 CHA versus an 8 CHA (for that instance). Advantage can overcome that stat modifier if the circumstance set up by the player attracts that ruling from the DM. That's a part of this edition of the game. Disadvantage swings it the other way.

  
  
  

  This answer is a challenge to the frame of your question, which has the embedded assumption that player skill should not to be taken into account by the DM. It is fine to do that, and in the end the DM is the one who makes the ruling on the outcome of a player's decision, action, or verbal declaration/saying, or if a situation accrues advantage / disadvantage.

  
  
  

  Bottom line: the other player is not a problem player, unless they are being a spotlight hog and not giving you a chance to shine. That requires some player to player communication and teamwork to sort out. If that player generally hogs the spotlight, then as a fellow player remind them that you get a turn also. Don't dump that on the DM; you discuss it with the player and you reach an accord with them. You both (your characters) are in this adventure together: work as a team.

  
  
  

  As a practical step, the next time there is a social encounter, volunteer to go first. If need be, tap that player on the arm and say "I've got this one" and do your best silver-tongued-devil thing.

  
  

You are not the DM

Each of us who has DM'd, but is in the role of a player, needs to address that our role is different as a player. We need to make peace with the instinct to try and back seat DM. (I still struggle with this some times). With DM experience, each of us knows how we would handle a given case as a DM.

Recommendation:

1. You are a player: play your character as best you can.




2. Enjoy the success of the other players, rather than seeing their good play through a competitive lens.



3. Work with your fellow players to form a better team.

An at table example of Charisma, low and high ...

I was recently advised by my DM that my ranger, who has the lowest Charisma in the group, and whom I have deliberately kept out of a couple of social interactions in favor of higher Charisma players (once with disastrous results when the dice went very cold - heh-heh, the dice are fickle!) is seen as the group's leader. In character, that makes some sense since he's often made suggestions that the group adopts (we generally make decisions by consensus), we are in a forest and he's a ranger, and he has a relative in-game who is under a curse that he's trying to lift. He has a motive to try and get to an objective. If we were to confine ourselves to the highest Charisma character being "the leader" it would be socially dysfunctional (as players) to try and force that player into that role if he doesn't want it.

The players form a small social group and thus the party's reactions will frequently reflect the real world, rather than the in-fiction, social dynamic.

When I talk about making decisions by consensus I take it OOC as well as IC: I ask my fellow players for input on "what spell should I choose that will help the party the most?" when I go up in level. Ranger spell picks are limited. I want their input because I want the team to succeed both in-character - the ranger's relative, the ranger's goals - and out-of-character, as a player. Those two states are not contradictory.

¹ The rules on Ability Checks state:

The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.

If the suggestion/persuasion seems to the DM to remove the risk of failure, then no die roll is necessary.

² For further reading on how some ability scores are problematic, this rant (caution, language) is worth a look.

I think this is mostly a DM problem, albeit a common one. With the physical stats it's very clear: a player can describe how his character moves the huge boulder out of the way, but with a -1 strength modifier it isn't happening. A character can describe how her character throws a stone against a lever from 100 meters away, but with a -1 dex score it isn't happening. Yet with charisma DM's are often confused. This is probably because they do want to encourage role playing.

Of course it's better for immersion if the PLAYER makes a nice attempt at a convincing persuasion, rather than saying 'I persuade the guard to let us through'. But this shouldn't supersede the stats of the character. And if the player does NOT come up with a nice story, the stats should still count. I've seen DM's change the DC in case of a good story by the player, but I'm not even sure how I feel about that, unless they do the same when the player describes how for example he puts some sand on his hands for grip, then puts his hand under the boulder and attempts to move it. There shouldn't be an artificial difference between stats just because one of them lends itself more to displaying it gathered around a table.

And when the smooth-talking player fails his charisma check because it’s so low, that too makes for fun role playing experiences. There can be situations where the NPC just laughs off some attempt that is now perceived as silly due to the low rolls, or just plain gets hostile. This doesn’t punish the player that chose a low charisma score either, it still provides for a story and would be more realistic in what would actually happen if he tried that.

In the end, I feel that too many people get stuck on the fact that ‘numbers/stats aren’t roleplay-y enough’. D&D models a world based on stats, and is (sort of) balanced around that. This may feel unrealistic to some, but the stats are just a much simpler approximation of how the real world also works. D&D has decades of experience of trying to balance characters and races against each other so that nothing is obviously much better all-round. If a DM then chooses to ignore this, it just means you’re not playing D&D but a more unbalanced system loosely based on that, where now a Fighter/Wizard with charisma as a dump stat now ALSO can talk their way out of everything.

This answer to a related Call of Cthulhu question is also excellent.

I'd talk to your DM about this.

I think the key observation is that low charisma does not necessarily equate to lack of eloquence. In fact, it can also mean the opposite.

For example, an eloquent sycophant or aspiring demagogue can still be highly uncharismatic and unconvincing for reasons that have absolutely nothing to with their ability to speak the – seemingly – right words.

Therefore, by far the easiest way to handle player's eloquence in case of a low charisma character would be to treat everything the player says as suspect: There is just something about the character that instills distrust whenever he or she starts talking. Sure, they can still convince some, perhaps many, but not all and definitely not all the time. In game terms, have them roll or even outright fail, if doing so would present the character better.

It would perhaps also help to consider the opposite: A high CHA character does not need to be eloquent.

The DM can award Inspiration for good role play.

Lots of great answers but I didn't see anyone mention Inspiration.

Awarding Inspiration is an effective way to encourage role playing and risk taking ... the character can have no more than one Inspiration at a time. (DMG, p. 240-241)

If the player did an over-the-top job of smooth talking, the DM could give Inspiration for his effort while still calling for a Charisma-based check. Obviously this is subjective regarding how the DM felt he did and what the situation calls for. It's a way to encourage his style of play while accounting for the ability scores of his character.

As the result of a charisma check is dependent on external factors at least as much as the words themselves, a failed charisma (skill) check can be explained in many ways. A few examples:

• "You are great with words sir, but I don't take advice from insert player race"


• "Nice try, but I just don't care"


• "You come in here, fully armed, and expect me to reason?"


• ...

almost every situation that should have called for a Charisma check
with his character was resolved by his own natural ability to charm,
sweet-talk, and work his way out of nearly every situation, not his
PC's, without ever needing to roll a Charisma check.

That's what advantage and disadvantage is for.

• When the player made a really convincing point, then they should roll their charisma check with advantage.


• When the line of reasoning the player came up with is flawed (but not so flawed that it's completely ludicrous from the point of view of the NPC), then they should roll with disadvantage.

The usual rule for skill checks applies, of course. Only roll a skill check if both success and failure have relevant consequences and neither success nor failure would break the campaign.

In my experience, roleplay must always be able to alter (mental) stats. Many old RPGs tend to lack flexibility in their rules, throwing lots of tables and numbers at you as you play unnecessarily hampering the flow of the game, so I have always changed the rules of whatever RPG I was playing on the fly as I saw fit in order to offer my players the smoothest and most realistic and fun gaming experience I was able to.

That doesn't mean that I dismiss mental PC stats over the skill of the player. Those are still the base of whatever may be happening, but I take into account how the player roleplays and grant advantages (and disadvantages) accordingly. I also set some kind of soft limitations: for example, an 8 CHA PC will NEVER smooth talk a guard after being caught on the scene of the crime he has just commited no matter how good is the silver tongue of the player, but maybe he could persuade said guard to accept a bribe, specially if the crime commited is just a small robbery. That way you are guarded against players trying to abuse their IRL skills in the game. A rule of thumb would be that a good acting by the player would ease the difficulty of a dice throw by a factor of one, and the opposite for a bad acting. Of course, most of the players won't be real actors, so I always offer the chance to not roleplaying that particular scene, not granting any bonuses but being safe from any maluses.

Then again, that's just me, and I'm sure to tell my players how I like to do things before the game starts, so no surprises for them and, to this date, the reception has always been a positive one. Not only do they have the chance of getting out of a situation relatively risk-free or with a small bonus, but they also learn how to RP in the process. I encourage them to roleplay, not just dumb throwing dices and checking tables, and when they do they feel truly rewarded. Sometimes, to that effect, I even grant them some "special points" in the form of automatic successes that can be expended at will, no matter if they are included or not in the game rules. Just notice, though, that some fresh players can become easily overwhelmed over the idea of having to "act", and that's the main reason why I don't force them to use to this system: if they feel confident enough they will try, and if they don't they just face a regular dice throw.

So, my advice would be to allow some degree of flexibility by the DM, but establishing some limitations to avoid abuse.

Question: How should a DM resolve a smooth-talking player with a weak Charisma score PC?

Answer: Exactly the same way they would resolve a gruff-talking player with a strong Charisma score PC.

The key is that the player is actually telling the DM the intent of the character and no matter how smooth or gruff they are in real life or how they stumble in the description, it is the character's Charisma that is used to resolve the event.

The DM might give the player a bonus for role-playing well or for bringing in plot points or contextual information or perhaps for just being funny. This bonus might manifest as a +1, as advantage or, perhaps best, in a change to the nature of the response of the NPC.

Whatever, the DM should tell the group what the happens based on how they judge the character behaved, not the player. This allows "gruff" people to successfully play high charisma characters and vice versa. Each DM will have a different style, but basically the DM should describe what really happens, telling the story in terms of the characters rather than their players.

Any other approach means we have to play ourselves.

It is the same for other ability scores too. It means unwise people can play clerics if they want to and normal people can play supra-genius wizards, for instance.

If people really try and role-play their characters then the DM's description of what happens will not be very different from player's expectations, and in fact in my experience it all rolls along nicely without the DM having to modify anything about the characters behaviour significantly.

However if the DM keeps on having to make big changes to how a character behaves compared to what player is saying, then perhaps the player should ask themselves if they are actually role-playing the character very well?

It is the same for Intelligence and Wisdom, particularly in my experience when a player keeps trying to make a low Intelligence character behave in clever ways.

TL;DR: the DM should tell the story of what actually happens to the characters, absolutely based on the player's input, while taking account of good role-playing and modifying poorly judged role-playing, so that everyone ends up with a fun believable tale.

I've a feeling this isn't as difficult a problem as you think to resolve. The real question is, should you resolve it?

So whenever he tries his smooth sweet-talking, if he fails his Charisma check, then the NPCs see through his flowery language and are suspicious or they think he is being patronising or tricky or something. His character just isn't good at sounding genuine to these people, or perhaps he makes an insulting faux-pas without realising it.

However, don't use this all the time. What he is doing is role-playing and he obviously enjoys playing this way, and it is nice to reward the effort with success. Whether it's role-playing according to his character is another point, and maybe the GM should have a word with him and suggest that if he wants to play this way he should swap a few points on another stat for Charisma or something.

I use three points to solve this matter:

1. 
   

   In my opinion a few dirty secrets or a silver tongue are a tool (an awesome idea is, too). They gain you as much leverage against a guard as a crowbar against a boulder. And thus, I give advantages for using them, if they seem appropriate. The bonus depends on how good an idea is. If you are simply saying that you have proof for a guards misdeeds, you have to be confident. And thus the success depends on a simple roll and the background of the NPC. If the player has a clever idea or information or an appropriate tool, you give them a bonus.

   
   
   

   • Pros: The players get benefits for good role playing. This encourages to role play and rewards good ideas, which furthers immersion. and that brings fun.

   
   

   • Cons: I don't see any. It might only be a problem when you play a game with a person that strictly holds to the rules and has a different understanding.

   
   


2. 
   

   A clever idea, a good tool and wise words aren't simply lying around. You have to have them. Tools can be found in chests. Ideas and words can be found in the character's brain. If a eloquent/smart player of a low charisma/intelligence character wants to use words or an idea, not befitting of his stats, just let him roll for it. Dump people have good ideas from time to time, too. It's just less likely. If the PC succeeds, he/she may use the player's ideas/words.

   
   
   
   • 
     

     Pros: You take player skill and character stats into account. And I find it rather logical. E.g.: Once I watched a RPG on YouTube and the party got sucked into an NPC's psychotic mind. At some point they had no idea what's going on, but I could understand the situation, because I read a manga with a mentally ill character, with the exact same illness just two months ago. I might not have extremely high stats in psychological knowledge, but I still succeeded to know this.

     
     
     

     Other example: I'm not very charismatic or eloquent. Some time ago I wanted to compliment a friend with a little joke. It sounded really dump to me, after I said it, but my friend was really happy and this I guess it was a success.

     
     
   
   

   • Cons: one extra roll and some players might pout, because they aren't allowed to use their extremely useful idea. But that can be helped by letting the other PCs help. The dump one could still give hints, as he has a kind of idea what to do/say, but isn't smart/eloquent enough to give it a concrete form.

   
   


3. 
   

   Sometimes, you can use following method:

   
   
   
   1. let the players say what they want to try e.g. "I try to persuade the guard to let me go."
   
   
   2. Let them roll normally, if they don't give further specifics.
   
   
   3. Let them roleplay the outcome.
   
   

    

   
   
   

   • Pros: you play By the rules and you can still roleplay. Can safe time by not having the player explain exactly what he wants to do, even if he doesn't even come to attempt to use his idea.

   
   

   • Cons(?): you stop the PC completely from using the player skill for this short situation. And it can hinder immersion as you don't follow the simple conversation, but stop to say what you want instead first. (This does only apply when the GM directly understands what the player wants. )

   
   

Example:

My kinda trouserless ranger, surprises the looting gentleman with a suitcase full of grenades.

Me: I try to calmly greet him and convey, that I am a sane human (because outside, it's teeming with zombies) and not evil, as he isn't looting my stuff.

GM: roll Charisma

I roll a crit. fail

GM: Critical fail. What do you say?

Me: BUUUUUUUUURP!!!!

The gentleman flees and falls down balcony.

As a DM, I'm often facing these situations as I'm currently running a campaign with an exuberant group who try to outsmart any situations.

In my way of DMing, I generally call for a dice roll against the NPC Charisma roll. But I'm considering the roleplay of the scene: if the Player convinces me as a person, I'll be a lot more flexible regarding the Player's throw and mine.
For instance, if the Player silvertongues but fail the dice roll with a close score, I let it passes. And if he fails hard his throw, the NPC could always says like "Mmh you convinced me, but my boss will kill me if I let you pass and I can't let that happening, sorry man" or "Yeah, you're a smart one, but you can't jest with me, prepare to die !"

Plus, in case of a very weak stat, I sometimes grant a permanent +1 bonus for it when the Player uses it with great success (not necessarily a critical success).
E.g: Last session a PC with weak strength was successful in literally throwing at a stronger PC the NPC they wanted to save, to make the group escape easier. The move was smart, and successful, so I granted a +1 bonus to the player when "throwing away people" because I considered his PC to be more experienced and skillful in these particular situations.

Anyway, I agree with the other answers that Charisma is one of the stats you shouldn't let be ruled only by the dice and stats. It's the DM art at its finest there.

So... I've been the player being talked about a few times in my life. I just ADORE being "clever", of doing something totally off the wall that totally works. I can't help it, its just too much fun. A while back, during a mod the other PC got captured and was going to be executed. I did some info gathering, gathered my resources, and proceeded to waltz in, save the girl, zip line to safety, and never even got spotted by the guards. Under the "only roll if there is a chance of failure" I rolled only once in the entire thing it was planned that well.

Afterwords, (once he no longer wanted to beat me with the gaming table) the GM told me "Your way to smart for playing a Int 10 character, I'm bumping you to Int 14, you probably should focus on raising it higher then that." In short, he complimented me on my cleverness, but reminded me that there was a distinct difference between my character's stats, and how I was playing him. Because I was having fun being "clever", I worked with the DM to adjust my stats so they better represented what kind of character I wanted to play, and the game moved on.

I would do the same for this player. Complement him on his verbal skills, and offer to allow his stats to get moved around if he is playing significantly more charismatic then the stats would suggest.