I put food out little at a time and many times a day....I try very hard not to have any left overnight. I do get the odd mouse but can live with that. Cat or no cat, where there is food, mice (and rats) will come. All a cat will do is kill some of the mice and also maybe some of the birds. If you want to discourage mice generally, then keep your garden tidy so little or no shelter or places to hide and remove as many food sources as you can. Sadly such a garden will be less hospitable to wildlife in general. As I said, I have struck a balance. Feeders are very near the house so lots of movement and disturbance which the birds are ok with. Rubbish is kept covered and away from the house, no compost heaps, bird food put out in small quantities and not left out overnight.