If the cats aren't around, flies will go after cat food. No fly stands a chance in our household though - the flies don't go after the cats because the cats go after the flies.
Flies are one of the only things that spur Oz into action, he doesn't bother with toys, but let a fly come into the room and his head is darting around watching it till it comes close. We actually saw him catch and eat one once.
They do go for the food though. When we had an old cat who didn't eat a lot, if we didn't pick up uneaten food there's be fly eggs on it by the end of the day.
Maybe a younger more active cat would see them off.
It is strange how flies bother horses and dogs but not cats. Maybe they move too fast? Flies here will go for the food if it is left around. But the cats will get a fly eventually, wherever it is! What I have a real problem with is ants - really tiny ones that I can hardly see if I am standing up. If I see a cat sitting with its head on one side staring at the floor, I pretty much know we have ants again.