One more thing, you dont need the pump yet. I have all the mods you do plus a cam and mine is fine without a fuel pump. I have heard it isn needed until up in the 12hp+ range.
You do know the
Edit button allows you to
edit your post, so you don't have to quadruple post, right?
Anyway, you've got a Predator? I have the 420 and I love it.
Now then, you said you need to cup your hand over the intake to get it started, right? I had a problem like this on another engine, when I was younger, where I used to have to push the butterfly open with my toe to start it cold. The point is your idle sounds like it may be low. If you look at the carburetor's throttle butterfly (where the linkage connects), and you'll see a black screw with a much larger screw head than the other ones. Turn this in about 1/2 a turn and try it. It shouldn't be out of adjustment, but you never know.
As for having to cup your hand, that sounds like your main jet is restricted -
Scratch everything I said.. More than likely, I know what your problem is. This may only apply to the bigger models, like I have, but it's worth a go. A few months ago, I was repairing someone's generator, which had a stripped jet and required a new carburetor. Little did I know, I ordered a carburetor that fit the engine, but it was designed for an engine with a smaller displacement. The result was a jet which was not designed for that displacement (389cc). The jet was most likely designed for the GX340, which shares many parts with the 390. Once I put the carburetor from my Predator 420 (It's really a 389cc engine) on the generator, it ran fine.
I'm guessing your jet is providing a fuel/air mixture that is not suitable for your displacement. The ideal ratio of gasoline to air, in terms of combustion, is 14.5. I would definitely look in to the modifications you made to the fuel system, since I seriously doubt your carb got gummed up in the past few days. As long as you have a steady stream of gasoline going to the bowl, you should be fine.