Fuel pump ?

It sounds like it's picking up debris in the tank and plugging the pickup tube.  Whether it be the sock or dirt.  After the 3-4 hours the obstruction drops back out.  I had an old tractor that did this once.  It had an oil bath air filter.  Come to find out it had never been cleaned before I got it.  it would run 10-15 minutes like a champ then die.  I thought it was the fuel pump  i tried everything then i took the boot to the air filter loose for something and started it up and it ran like a champ.    That's how i found out about the oil bath filter.  but it does sound like you have an obstruction somewhere.  Good luck!

Yeah thanks for the feedback unfortunately it isn't a tank   

You're definitely right that it's most likely not the relay but I would definitely check all the easier stuff before dropping the tank. Personally I would see if I can get the meter hooked up as close as possible to or all the way back at the tank. Then take a voltage reading when you have 32 PSI and when you have 10 PSI. Alternatively find a way to provide supplemental 12v power to the fuel pump. I'm betting the voltage doesn't change regardless of the fuel pressure in the system. 

You say that the pressure drops from 32 to 10 PSI under acceleration. Is that under any acceleration or just heavy acceleration? 

You said if you shut it off for 3-4 hours it behaves the same as above. Have you tried waiting until it's at 10 PSI then turn the motor off and right back on? Does the pressure revert back to 32 PSI or is it still at 10 PSI? does the system need 3-4 hours to regain the 32 PSI?

Ultimately if I had to hazard a guess I'd say your fuel pump is on it's way out. Or at the very least the pump pickup sock in the tank is plugged up. They are available on Amazon.