To recoup is a kind of recovery: If you lost some money but then made that amount back, you recouped your loss.
When you recuperate, you get better after being sick. And when you recoup something, you get better or bounce back after a loss. Businesses that lose money try to recoup it by throwing a sale or cutting their budget. If a runner falls behind in a race but then speeds up to the front of the pack, he's recouped his lead. In some cases this word also means "to reimburse."