What is the correct usage of the verb recoup with preposition for or without?
Please support your answer with examples.

Andriy - do you have a particular question in mind? There are lots of ways to use any given word and it would take a long time to write a lesson about all of the possible usages of one word, including examples. For example, sometimes you do use "for" with recoup and sometimes you don't - most of the time you don't, but it depends on the structure of the specific sentence.

I recouped my losses (the meaning here is to regain or make up for something lost)
It is possible to recoup someone for his losses (the meaning here is to reimburse or compensate)

The typical scenario is the first one, in which we don't use "for" - if you have a particular sentence (and it isn't from one of our banned sources) please post it. Generally speaking, if you're taking the test and aren't sure - guess the option that doesn't use "for."