Perhaps the fuel pump motor is wearing the commutator down to the point that the brushes are arcing or producing a back-EMF to fry the contacts on the relay. It could be loading the circuit down to the point that pump stalls on the second try. The pumps usually start when they are cold, but not after running for a while, if they are malfunctioning. That's what mine did.
It might not be the relay, but the pump itself, since the primary circuit of the coil is activating and you hear the click. But the pump is not running, and you don't hear the buzz-click after 2 seconds.
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Pete
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