Umpteenth time today, but basic sanitation by locking up the bulk food and the food in the coop will send the rats and mice elsewhere. Trying to trap and poison your way out of a rodent infestation doesn't work. Each week someone posts another rat and chicken thread rather than do a search to find where the topic has been discussed for months, sometimes years, and all the solutions laid out. Look over in Feeding and Watering Your Flock or just do a search of the entire forum for something like rat chicken or rat proof treadle feeder.
You are going to HAVE to by a treadle feeder with a spring loaded door, a distant treadle, and a counterweight, nothing else is going to work if you are already infested. If you are not already infested one of the pretty feeders like the Grandpa or Feed o Matic will work till the mice start pushing their way into the feeder. Hanging feeders do not work either. Trigger feeders do not work. PVC elbows and buckets do not work. All will work before the mice and rats find you but not afterward.
You can feed twice a day by hand but the rats and mice have zero problems feeding during daylight in a protected coop safe from predators. Your egg production will go down by restricting feed as a hen is always racing to digest enough protein to lay eggs. Ditto on broilers raised for meat.
Cats are either mousers or not and many will not tackle a rat. If you dump a cat outside it is going to devastate the wild bird and rabbit populations after it runs out of rats. A cat might help if you are rural or in an urban area with a good rat population.
If you do go with a treadle feeder watch the negative reviews very carefully. You can discount some of the "my chickens won't use the feeder" posts, those are people that won't follow instructions. Look for the reviews that show the rats getting into the feeder.