What the question is asking is how the cat’s owner can stop his or her cat behaving naturally. It is not possible to do that if mice are available. It is not even kind to stop a cat acting naturally. Hunting is at the center of a domestic cats psyche. And it pleases a cat to bring mice home to train her human companion/kitten!
The only solution, as I see it, is to ensure that there are no mice to hunt. If a cat is an indoor/outdoor cat, free to roam, then you cannot stop your cat hunting mice from time to time and bringing them inside.
Although it does depend upon the individual cat. I once owned a female cat who throughout her entire 18 year life only brought into the home one mouse and she did not eat it. There is a solution only you might not know your cats interest in hunting until you have adopted her.
Today I live with a male, tabby cat, who was raised as a feral kitten and he brings in pretty well anything he can get his hands on.
However, he is a confined cat. He lives in my house and in my back garden (yard), which is bounded by a specially designed cat confinement fence. There are no mice in my back garden…fortunately. And therefore he has never caught a mouse. He has caught three pigeons but I stopped him bringing them into the house because he hunted them during the daytime. I am not saying that he will never bring a mouse into the home but it’s less likely under this form of cat guardianship.
The point I’m getting at is that if you want to stop your cat bringing mice into the home then you have to confine your cat to a highly enriched (from the cat’s perspective) backyard where it is unlikely mice will nest and breed. That may be somewhat impractical but I think it is a reasonable suggestion.
There is a fancy suggestion on Quora.com about a cat flap which employs facial recognition to let the cat inside. When a cat has a mouse in his jaws the device recognizes it and bars entry. However, as far as I’m aware, this device is no longer available!
If my suggestion about confining your cat is adopted then it is unwise to eradicate mice from the backyard with poisons. If you do so you will poison your cat. In fact, I would not eradicate mice in the backyard at all. It would be cruel to do so. However, if it is configured in a certain way it should minimize the possibility of mice nesting and breeding.
I welcome better ideas!