To be honest from what he describes, I don't think its the accelerator pump. The accelerator pump only does something when you step on the gas harder. It squirts a little fuel in to the carb which gives the carb a change to see an increase in fuel demanded and compensate. Thats whats nice about fuel injection, the response to more fuel is almost immediate, while a carb takes a second to catch up. The accelerator pump is there to compensate for this delay. Usually an accelerator pump problem would be when you go to pull out into traffic the vehicle hesitates badly.
There are so many things that could be wrong, it could be ignition, it could be a vacuum leak, it could be the fuel pump, it could be..........................