Put an animal, a bird, etc. out of its misery
Definition 1 of 2
Pro Tip 1/3Compassionate Context
Used when an animal's suffering is severe and cannot be healed, implying humane and caring action. 
After the accident, it was kinder to put the horse out of its misery.
Pro Tip 2/3Terminal Illness
Often relates to animals with incurable diseases or irreversible injuries. 
The cat was suffering from terminal cancer, so we put her out of her misery.
Pro Tip 3/3Last Resort
It's considered an act of last resort, meaning all other options to save or relieve the animal have failed. 
The vet suggested surgery, but if that failed, we'd have to put our dog out of its misery.