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In Spanish we use this expression abuse of your trust usually in formal writing to apologise in advance when we want to ask for something that one wouldn't do if wasn't because our trust.

Example.

Sorry for making abuse of your trust but I wonder you'd do a favour. Can I borrow your car for the weekend?

How would you express this in English?

Thanks.

The phrase makes sense in English but the meaning it conveys is far too severe for the context you give. An "abuse of trust" in English would be a very serious matter and even describes some illegal acts - for example, the definition "abuse of a position of trust" is used in British law to describe sexual activity between a teacher and student, or a caregiver and care-recipient. Not quite the same thing as asking to borrow someone's car.

I think the equivalent expression you are looking for is:

Sorry for the imposition...

Preceding a polite request this normally implies that you are aware you are asking for slightly more than would normally be expected.

Another alternative may be to say:

I hope this isn't presumptuous, but...

This is another way that some may soften a request by apologising in advance for presuming that it is possible.

English uses the phrase abuse of trust as well, but it would not be used in the kind of sentence you proposed.

Here are some examples of abuse of trust taken from English-language periodicals (compiled by The Free Dictionary)

"Your actions in pursuing an emotional and sexual relationship, and in having sexual intercourse with [a patient receiving psychiatric care from you], were a gross abuse of trust."

"And a new offence, abuse of trust, will stop adults exploiting under- 18s in schools, the forces and places of care."

"In a gross abuse of trust, Pringle, one of a small number of people with access to the committee room, began taking cashed tickets and re-cashing them for himself."

The last example is perhaps the easiest to explain. Pringle was given access to the committee room, where something valuable was stored, because people trusted him. They trusted him not to steal the valuables. He abused their trust by stealing the tickets.

Asking for a favor from someone may be an abuse of trust, but probably not in the case you described.

If you convince someone to let you borrow their car by lying to them, that may be an abuse of trust. If you repeatedly ask to borrow someone's car, but are always honest about the reason, there is no abuse of trust, but it may be an abuse of their goodwill.

I don't mean to take advantage of you or our friendship, but could I borrow your car?

The Spanish expression used in everyday language really means: to take advantage of someone due to the trust they have in you.

It has nothing to do with the legal term, breach of trust. The idiomatic expression and legal expression are two different things in English.

No quiero abusar de tu confianza: I don't want to take advantage of you [in the sense of: given the trust you have in me]

As a native (Scottish) speaker, I would use something like:

"I'm sorry to impose"

or simply:

"I'm sorry, but I wonder if you would mind doing me a favour?"

As others have said, in English "abuse of trust" is a rather strong statement. This answer says that the expression better translates as "to take advantage of someone due to the trust they have in you." Have you considered an alternative like "abuse of your hospitality?" That seems a better translation of the sentiment. E.g.

Sorry to abuse your hospitality, but I wonder if you'd do me a favor. Can I borrow your car for the weekend?

This would sound fine to me in American English. It does suggest that you are a guest of the person of whom you are making the request. You can replace hospitality with other words, e.g. generosity would be more general. Or friendship (which would imply that the two of you are friends).

Another possibility would be "Would it be too much to ask?" E.g.

Would it be too much to ask if I could borrow your car for the weekend?

If the goal is to soften the request.