I presume the cat can come and go freely eg cat door or similar? I used to have this problem until I limited where the cat could enter and what it could access, once it entered the house - I changed her access point to a small window in what had been a built-on afterthought downstairs lavatory, and set it up as 'her' space to enter if she wanted to come in 'independently'. I made sure there was no egress point for even the smallest rodent - if you can stick a pencil in, a mouse can get in - put in a preferred piece of fleece on a cushion in a big box for the cat, and some of her favourite food, and a couple of cardboard boxes of different sizes but far too small for the cat to get in (to persuade any rodent to go into a box as the safest retreat spot.
Apart from anything else, having to jump - albeit only a few feet, and onto a very wide windowledge - with a mouse, or bird, in the mouth, was more difficult than coming straight in through a cat door, and I often found bits of mouse on the path outside where the cat had clearly decided to eat what bits she fancied rather than struggle with carrying the takeaway and risking dropping it. So fewer mice were being brought in in the first place.
When she did succeed in bringing a mouse in, it was enclosed within the four walls of the lavatory and when I went to check on the cat and let her into the house 'proper', I could check for rodents with little to no risk of any entering the house as any the cat had brought in live would be hiding in or behind one of the cardboard boxes I had placed for that purpose.
It took a few weeks of vigilance and plenty of her favourite food served ONLY in her 'private entrance parlour' to get her to 'convert' reliably to using the 'lavatory entrance' rather than her standing outside the back door miaowing at her now-closed cat door, but it worked fine, except occasionally in high summer when the back door was left open in good weather - but that was never the 'busy season' for mice anyway; autumn was the busy season.