The accl pump is there to help cover up a lean condition with quick movements of the throttle.
If you are having issues taking off at light throttle then I'd look at tuning that part of the system. If you have an APT carb turn the APT screw CCW about 1/8 turn or so and test drive. Continue doing this until it richens things up enough you don't get a stumble/hesitation, or bog right off idle.
It may also help to fatten up the idle system some to put more fuel to the transfer slots. \
I've cured issues just like you are having by opening up the DCR's a couple thousanths after not getting the results I wanted with the APT system or slightly smaller primary metering rods.
I'm also curious as to what jets and metering rods you are using (primary side), and what engine the carb (specs) it's being used on?........Cliff