Fuel Pump question


If my 52 wagon sits long enough that the carburetor runs dry, I can't get the fuel pump to fill the carburetor bowl.  The starter motor doesn't spin fast enough for it to pump fuel.  Is this a sign that the fuel pump is not up to snuff?  It isn't a big deal to prime the carb , but I also have a slow acceleration situation, and I'm wondering if I should suspect the fuel pump.

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