You may have multiple issues.
The hard starting issue is probably the fuel:
http://www.thecarburetorshop.com/Troubleshooting.htm#HardstartcoldThe hesitation is probably a different issue. Before throwing rocks at the carb, I would check the dwell, timing, and distributor advance mechanisms.
The most common issue with 1937 and newer Carter W-1's is not the accelerator pump, but it is related. There is a bracket attached to the back side of the throttle shaft that is attached to a rod that activates the pump and the metering rod. The round hole in the bracket wears oblong, and the round rod inserted into the hole wears with a flat side. When moving the throttle, the throttle can rotate significantly BEFORE the accelerator pump actually moves. Replacing a good accelerator pump with a new accelerator pump if the bracket/rod combination is worn only lightens your wallet.
Very easy to test the above.
Remove the dust cover
Watch the accelerator pump, and rotate the throttle
The accelerator pump should move almost instantly
If not, pull the pin spring holding the rod to the bracket, and inspect.
While these parts are not in modern rebuilding kits, they are available mail order.
The vehicle CAN be driven with a defective (or missing) accelerator pump. I have posted this before, but again: it was common practice during WWII (at least in rural Missouri) to REMOVE the accelerator pump ENTIRELY (gasoline was rationed). Many farmers survived driving this way for several years.
Jon.