All of our animals have chips just in case they get lost.
Take it somewhere to see if it has one. The humane society or even a vet would probably scan it for free. It's got a collar so it's somebodies' cat. If it's losing weight, I'd say it'd be pretty strange that it's leaving dead food on your doorstep. It sounds like it's lost but good, because unless someone in your house has fed it, it wouldn't come back.
If it doesn't have a chip then, I'd try the hose idea, or just keep a bucket of water near the door. Let it go bother somebody else.