A word for "mercy killing"
Solution 1:
StuartLC beat me to it. :(
The word for mercy killing is euthanasia.
Usage:
The veterinary team could not heal the dog; it had to be euthanized.
Many religious people take issue with euthanasia as being no different than any other form of murder.
Solution 2:
In addition to the already mentioned euthanasia another phrase (not a single word) that is used, most often in the context of war, is Coup de grâce from the French "blow of mercy".