Pumping the gas on a fuel-injected car does absolutely nothing, as there is no accelerator pump. Holding it down fools the MAP sensor into a faulty reading when starting the car. How does it run when it finally starts hot?
It does sound fuel-related if holding the pedal down helps it start. Too much fuel (rich) needs the extra air provided by an open throttle to combust. Diagnose for a fuel hot-soak issue.
An injector that drips after shutoff is possible. A fuel regulator problem is possible (check fuel pressure). Vapor lock is possible if pressure in the fuel line drops after shutoff.
Pull the doghouse and air cleaner for a look. Fuel pressure is checked by tee-ing in a fuel pressure gauge.