Do cats bring gifts according to their power?

No, cats bring gifts depend on the items you placed, the food and the most important, the time their stay on your yard. The time could be more or less by the food or the items. Also, there are specific cats that only appear with a certain items + food. i.e Tubbs only appear if you have food that is not the "free food", he will come and eat the entire bowl and stay for a short time.

You can do a test, puts some food, see what cats come first, wait atleast 20+ minutes more and then see what cats came after the first one. Next quit all items. Doing this, all the cats leaves and you can see how amount of fish the cats give you.