Etymology
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recoup (v.)

1620s, in legal language, "to deduct, keep back as a set-off or discount," from French recouper "to cut back" (12c.), from Old French re- "back" (see re-) + couper "to cut," from coup "a blow" (see coup). The sense of "to recompense for loss or expense" is from 1660s; the meaning "return or bring in an amount equal to" is by 1860. Related: Recouped; recouping; recoupment.

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Definitions of recoup

recoup (v.)
reimburse or compensate (someone), as for a loss;
Synonyms: reimburse
recoup (v.)
regain or make up for;
Synonyms: recover / recuperate
recoup (v.)
retain and refrain from disbursing; of payments;
Synonyms: withhold / deduct
From wordnet.princeton.edu