That's a common and affordable pump. I'd begin with that. Maybe a carb kit. I would imagine your fp is similar to this one though it might be plastic. They come in a couple of different shapes and some are rebuildable and some are throw aways. The one in the link is rebuildable.
Fuel Pump Kit Kohler 24 393 04 24 393 16S Briggs Stratton 491922 10876 | eBay