Cats usually only eat fresh prey which they just have killed. They aren't scavengers.

They easily throw up because in case they should have eaten something that is bad for them they can get rid of it easily.

Dogs are different. And wolves also transport food to their little ones which they regurgitate.

So most of the time a cat won't eat the thrown up food.

However that mechanism of throwing up is also started when a cat eats too fast and too much. My Tigris is known for that. So, 1 minute after eating he sometimes has the food up again. It looks fresh and I am sure it tastes fresh. Tigris however only eats it in very rare cases. Most of the time he leaves it to us to clean up the mess

So if it is only slightly used food I can't see a problem in eating it. But if it's yucky half digested stuff I would clean it away before he eats it again.