You can put the animal out of its misery .
It means to euthanize.
I suppose you could say the same of a failing business enterprise, figuratively, and even of a dysfunctional relationship, when speaking with a sort of grim humor, where you're casting the relationship as a badly injured or terminally ill and suffering creature.
ell.stackexchange.com/users/10264/Tᴚoɯɐuo
Put to sleep and put down are common euphemisms for euthanizing an animal - "put Fluffy to sleep", or "have Fluffy put to sleep", or "have Fluffy put down.
"
I've heard both used in other contexts such as bad relationships or failing businesses.
ell.stackexchange.com/users/45034/John Bode
A coup de grâce is a death blow to end the suffering of a severely wounded person or animal.(Wiki)
ell.stackexchange.com/users/84814/user84814
I'd use take it behind the barn and shoot it for putting an early end to something instead of having it drag on.
A similar saying is take it out back .
ell.stackexchange.com/users/84776/Nathan
Tᴚoɯɐuo gave the best answer for killing an animal mercifully.
In the case of a business or investment, while we might use that as a metaphor, some common phrases are cutting your losses and don't throw good money after bad .
ell.stackexchange.com/users/6111/Barmar
'It would be better off dead', 'I'd be better off dead!'
is a common UK and US idiom, in speech and writing.
It sounds very strong unless we are already talking empathically about death.
'Flogging a dead horse' ie continuing with a hopeless endeavour.
'
Hastening the inevitable' clichéd but appropriate.
ell.stackexchange.com/users/84836/OookLout
" pull the plug " Literally, this means to pull the plug of life support equipment and let someone die.
Or less morbidly, unplug a bathtub and let the water drain out, ending the bath.
Metaphorically, it means to put an end to something:
We've been more like roommate than lovers for the past 6 months.
It's time to pull the plug on this relationship.
I hate to pull the plug on this party, but it's midnight.
I've gotta get some sleep.
There's also " pull the trigger " which has multiple connotations depending on context, but the same basic meaning of ending a period of waiting: Pull the trigger to kill/stop something
"She's suffered long enough.
You've just gotta pull the trigger and get it over with."
Pull the trigger to initiate/start something (like a starter pistol at a race)
"We're ready--just waiting on the boss to pull the trigger so we can start." Commit to a decision/action "Dude!
It's not that complicated!
Just pull the trigger and pick one!"
ell.stackexchange.com/users/12413/miltonaut
The thing in question can be a wounded animal or a dysfunctional relationship. In mild sense, it can be an object or idea that requires a lot of investment for it, and when it turns out that it not working you want to cut it down to avoid further loss.  Is there an idiom, proverb, or common saying that describes this situation?   Related:  Is there an idiomatic way to say "go to the path of no way out"?
Is there an idiom about how humanely killing something is better than letting it live in pain?
ell.stackexchange.com
<idiom-request><proverbs>
ell.stackexchange.com/questions/184925
ell.stackexchange.com/users/11458/Ooker
To put something out of its misery, to put something to sleep, or to pull the plug are suggested terms in this context. A  coup de grâce  is a death blow to end the suffering of a severely wounded person or animal. Flogging a dead horse means continuing with a hopeless endeavour.  Hastening the inevitable is also used. Putting an animal out of its misery, putting them to sleep or having them put down has this meaning, while 'pulling the plug' may refer to removing someone's life support. A coup de grâce is a final death blow to end the suffering of a severely wounded person or animal. 'Flogging a dead horse' suggests the futility of continuing a hopeless endeavour, while 'Hastening the inevitable' could be an appropriate idiom.
Putting an animal out of its misery, putting them to sleep or having them put down has this meaning, while 'pulling the plug' may refer to removing someone's life support. A coup de grâce is a final death blow to end the suffering of a severely wounded person or animal. Flogging a dead horse' suggests the futility of continuing a hopeless endeavour, while 'Hastening the inevitable' could be an appropriate idiom.