2008 Honda Accord Making Strange Sound?

My 2008 Honda Accord EX-L is making an odd noise. After I start the car up and drive for approximately.5 mile, it makes a soft/dull thud sound. The sound originates from the left rear fender area (just below the gas tank), its almost as it I ran over something, yet I haven't. After the car makes this single dull/thud sound, it ceases and does not make any other noises (except only after I park my car, shut the engine off, let it sit, and restart the car, at which point the dull/thud sound occurs). The car only makes this sound one time after I start the car, however it makes this sound EVERY TIME I turn it on and drive a short distance.
I suspect a faulty or failing fuel pump?
Please help and any worthwhile response is truly appreciated!

Might be the exhaust system that has a broken mount. When starting the car the motor may shake a bit more as it cranks and starts causing the thump as the exhaust pipe hits the body.

A fuel pump makes no noise before it goes out (aside from a slight whirring sound that says it is pumping fuel), they are silent killers as far as when they decide to take a dump.

Go to your mechanic and ask him/her.

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