An accelerator pump is a mechanical (not electrical not pneumatic) device that squirts extra fuel into the carb to momentarily richen it to aid acceleration. It is usually a diaphragm or a plunger type device. It actuates when you open the throttle plate. A check valve allows gas flow into the pump but prevents it from returning to the float bowl so it exits into the carb thru a small orifice.
Whet you describe is a failed pump. The plunger should cause the engine to speed up a bit unless too much is going in, then it slows down.
My '89 Toyota 22R engine was a plunger (little piston) accelerator pump. Whereas Troll is working on an old carbed Toyota pickup, his is probably similar.