OK, getting somewhere..
The part marked "Main Nozzle" is what you are after.
That part you will see has a bunch of tiny holes.
Those holes clog, for no ryme or reason, they just clog from deposits build up.
That nozzle is actually how the fuel gets up into the throat of the carb.. The little pin holes in that part helps to break up the liquid fuel into fine particles.
Even if the holes look OK, run a pin or needle through each hole gently scrubbing the sides of the holes. This is a brass part so you want to be careful to not enlarge the holes. You also want to check that the hole on the end you can't see is open.
The bottom pix is the altitude adjustment, basically leans or richens the the fuel mix to compensate for the altitude you are at. Engines tend to run too rich at very high altitudes so the general fix is to change the jet to a smaller one. Your gen has an adjustable jet, if it is set for to high of altitude than what you are at it will run lean (stall or run rough)..
I think from the diagram it is basically a "needle tip" so removal may not be needed. Since you are not able to turn the screw, I suspect that the brass which the needle is made from may have reacted with the bowl material and froze in place.. Not sure if that is a problem or fixable without replacing the bowl..