1.
If you ~ something that you have been eating or drinking, you eat or drink the last part of it with the result that there is none left.
Kelly finished off his coffee...
He took the bottle from her hands and finished it off in one long swallow.
PHRASAL VERB: V P n (not pron), V n P
2.
If someone finishes off a person or thing that is already badly injured or damaged, they kill or destroy them.
They meant to finish her off, swiftly and without mercy.
PHRASAL VERB: V n P
3.
see finish 2