Distressed B&S...Racheal?

How to repair?

Hello i have a 3.5 B&S lawnmower engine that once primed with fuel will keep running, however once you shut it down it won't restart unless you physically prime the throat of the carb any ideas. It has a primer on it however doesn't seem to be doing much. Also if i try to start mower when it is hot it will, however after a few minutes of cooldown i am back at priming. Any help would be appreciated.

Racheal B