That would the fuel tower that contains the main jet and nozzle. The small cup like item is the fuel bowl that the fuel fills up so the main jet is submerged. What appears to warp around tower is removable screen that is there to filter out trash. As IL just posted the main could clogged but I have the internal passages of the tower also clogged.
BUt it sounds like the tank is pretty rusty and if so it is a lost cause as the tanks are no longer available and have been for quite some time. A shame in one sense as a lot of good equipment just get junked for the lack of a good fuel tank. I think part of the problem was that Briggs overpriced the tanks and people simply quit buying them.