A faulty fuel pump can contribute to a backfire on startup, but it's not the most common cause. A weak or failing fuel pump might not deliver enough fuel to properly support combustion. If the air/fuel mixture is too lean (not enough fuel), the engine can backfire, especially on startup when the engine is cold and needs a richer mixture. However, a backfire is more often associated with ignition system problems (spark plugs, wires, coil), intake leaks, or problems with the engine's timing.
So, while a faulty fuel pump is *possible*, it's more likely a symptom of a broader issue, and investigating other potential causes is crucial.