Acceleration problems

Hey guys,

I am having some issues with my car. When I floor the gas pedal, the car hesitates as if something is holding it back until I lightly release the pedal and then it drives just fine. On normal highway driving, there is no issue. Problem only arises whenever I stomp on the pedal.
Secondly, I am having hard starts especially when the car has been lying around for some time (like hours or a day). This is compounded by the fact that the fuel pump relays keep going bad at an alarming rate, which means that most of the time the car is without a functioning pump relay. There is also no whirring sound on the fuel pump when I turn the ignition on.
Kindly assist.