Before you waste you time chasing issues in the pump circuit, make sure you're working with the correct parts! Carter AFB carbs came with a dozen or so different accelerator pumps over the years. The cup diameter and overall length of the pumps vary. The carburetor number will tell you which pump is correct. If the number has been ground off your carburetor we have no way of knowing which pump is correct.
NOTE: Be sure choke is wide open so fast idle cam does not hold throttle valve open
NOTE: Be sure fast idle adjusting screw does not contact fast idle cam and hold throttle valves open.
With throttle connector rod in center hole (medium stroke) of pump arm, the distance from the top of the bowl cover to the top of the plunger shaft should be as specified on carburetor spec page.
Adjust by bending throttle connector rod at lower angle.
HESITATION
Hesitation can be caused by electrical (usually distributor), as well as by the carburetor accelerator pump circuit. We will not address the electrical system, that is outside our area of expertise.
1st do a quick accelerator pump test to determine if the pump circuit is working properly. If you don't get a good squirt when hand testing, you've got an issue in the pump circuit.
Additional Diagnosis in order of probability:
Tip - The AFB pump link, links the pump lever to the accelerator pump. When you look at the link installed from the front of the carburetor, it should look like a 'S'.
If you need a new pump link, purchase it here.