Discuss and learn French: French vocabulary, French grammar, French culture etc.
Well actually, "finish off" can sometimes have that meaning in English, usually when referring to a person on a battle field, or e.g. if the cause is an illness. For the example of an injured animal, you can talk about "putting an animal out of its misery".
In a fight, you might talk about "dealing the final blow" ("coup" in a fight is usually "blow").