Dissonant whispers (PHB, pg. 235) says:

The target must make a Wisdom saving throw. On a failed save, it takes 3d6 psychic damage and must immediately use its reaction to move as far as its speed allows away from you. The creature doesn't move into obviously dangerous ground.

If someone holds their action to cast this with the trigger being to cast it on a hidden enemy when they reveal themselves, then when the enemy does reveal themselves, and the readied spell is cast, and the enemy fails their save, they must then immediately use their reaction to run away as far as their speed.

However, since this is now happening on that enemy's turn, do they still have their movement for that turn, or did they just waste it running away as per the spell's effects?

In other words, enemy normally has 30 ft movement. If the spell forces them to run 30 ft. away during their turn, can they run back or do they now have no more movement left that turn?

The enemy will have used their movement.

The reason is fairly simple, given a clear look at how movement works: speed is defined as the amount of movement you may use during a turn (unless otherwise noted). Since they’ve used movement equal to their speed, they’ve reached the limit and can no longer move.

The way to think about movement and speed is that you count up as feet of movement are taken until speed feet moved.

The enemy cannot move any further that turn

Dissonant whispers says:

...use its reaction to move as far as its speed allows away from you.

The rules for movement say:

On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.

[...] However you're moving, you deduct the
distance of each part of your move from your speed until it is used up
or until you are done moving.

Each turn you can only move a distance up to your speed unless some other feature specifically says otherwise.

Dissonant whispers forces you to move but the movement is not indicated to be special in any way. The rules are clear and simple. Any time you move on your turn you use your movement to do so. The spell causes you to move, therefore it also causes you to use your movement when you do so on your turn.

The fact that it uses a reaction does not mean that this not normal movement, it is simply an added effect of the spell that adds a cost to the movement.

Normally, the spell's forced movement would take place on another creature's turn (since most of the time the spell is cast on the caster's turn). However, in the scenario above, the reaction does indeed take place on the moving creature's turn. As such, the movement counts against the maximum the creature can move in their turn like normal. There's nothing in the spell that indicates it acts any differently. Thus, the creature had no movement left at the end of their reaction.

They can still Dash

However it is worth noting that the creature could use the Dash action to give itself extra movement and use that to move wherever they wished.

No, the affected target can still move because the spell used its reaction, not its movement to move

On PHB pg. 189, under Your Turn, emphasis mine:

On your turn, you can move a distance up to your speed
and take one action. You decide whether to move first
or take your action first. Your speed—sometimes called
your walking speed—is noted on your character sheet.

On PHB pg. 190, under Reactions, emphasis mine:

Certain special abilities, spells, and situations allow
you to take a special action called a reaction. A reaction
is an instant response to a trigger of some kind, which
can occur on your turn or on someone else’s... If the reaction interrupts
another creature’s turn, that creature can continue its
turn right after the reaction.

So in your scenario the affected target would have used its reaction to move, but then it would still have its action, potential bonus action and movement to be used during its turn. As NautArch pointed out, the wording of the spell is specific in that it states:

...use its reaction to move as far as its speed allows away from you.

The spell says nothing about the creature using its movement for the turn, nor does it state that the creatures movement is impeded in any way. Since it doesn't, there is nothing preventing the creature from using its movement to return to where it was.

If it helps: There are 5 processes that a character can utilize per round. Action, Bonus Action, Movement, Reaction, and Interact With an Object (although this is a relatively minor action and is not paramount to this discussion). Whether you qualify for a bonus action or a reaction is dependent on circumstance, and if something uses one of those, they're independent of the other 4.

So for example: A Battlemaster Fighter using the Ready action to allow a Rogue to slip past an enemy on the Rogue's turn could state: I'm going to Ready my Maneuvering Strike and shout at the Rogue, "Run! I have him distracted!" The trigger will be as soon as the Rogue acknowledges the plan.

So now on the Rogue's turn, the Rogue says, "NOW!" The fighter triggers Ready Action, uses Maneuvering Strike and selects the Rogue as the recipient. The Rogue gets to move up to half their speed using their reaction, so the Rogue moves 15 feet, not triggering an opportunity attack from the enemy. Then the Rogue uses his movement to travel a further 30 feet past the enemy, Cunning Action Dash for another 30 feet, and Action Dash for yet another 30 feet, totalling 105 feet of movement away from the distracted foe.

Another example of this is the Rogue: Scout subclass. At 3rd level they get the Skirmisher ability:

Starting at 3rd level, you are difficult to pin down during a fight. You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you. This movement doesn’t provoke opportunity attacks.

And yet another is the College of Glamour Bard, using Mantle of Inspiration:

When a creature gains these temporary hit points, it can immediately use its reaction to move up to its speed, without provoking opportunity attacks.

I hope these further examples help illustrate that your reaction is not limited to mere opportunity attacks and can in fact also apply to movement.