how long after shut down? minutes or days? does the accelerator pump work once you get the engine started?
the accelerator pump needs gas in the bowl to work, if the carb is full and the pump doesn't 'squirt', it needs replacing, but in that case it's not likely to be working properly at any time - if the pump doesn't squirt and the carb is empty, you need to crank it awhile to get the gas recharged
Bill