What would be causing an intermittant starting problem with efi fuel prime pump not working?

If you are sure the fusel pump is losing pressure or stops working check the EFI relay in the fuse box under the hood, check to make sure the EFI fuse is not loose or cracked, locate the circuit opening relay it controls the fuel pump and try a new relay, if nothing else replace the fuel pump, another possibility is the air flow meter, there are contacts inside that turn the fuel pump on and off as the meter door opens and closes.

running the pump dry is bad. on all EFI cars
the pump is fuel cooled and fuel lubricated
if you run any pump dry, you risk pump damage
but lets hope it not damaged, hope is good.
key on 10 times, hear it run?(the pump) let it run 10 times not started
and pray it works, and this trick primes it and gets all air out of the line
the 3 second run time is called just that auto prime.
then run it.
never run EFI cars dry, its a rule.
the dash chime told you it was low, ignore that at your risk.

no start , crank perfectly
so , got spark?
the ECU has programming (ROM) and in fact kills all fueling with no spark , (even if it thinks there is no spark, even if spark is ok)
all ECU use a way to read back spark quality, (ALL){LAWS}
how honda does it in code, i don't know , see C R O M E forum)
turn the key over means nothing to me.?? sorry, it doent/
try saying what postion? cranking? or just keyon.
now some more EFI rules.
The EFI (some) do not activate the pump relay until cranking.
some do just key on for 3 seconds, read the FSM, its free.
need a link to the free manual? read the ECU logic on that.?
and prime has a special meaning, it means purge air,
its not for fuel. some system send a fast prime pulse. to clear air from the pressure lines, this plus does not inject.
that is how EFI works,

the flow
1: key on, ECU sends prime pulse, (air purge)
2: some ecu, run pump for 3 seconds, and pressure hits 35psi? RTM.
3: all ecu activate pump cranking and running.
4: now check pressure. !" 36 psi FSM stated spec"
5: now crank, if the ECU sees no spark errors, it injects
6: if the ECU sees WOT. it stops injection to clear a flood, cranking.
those rules cover most ECU made.

honda this year has a special feature called,
Fuel Cut-Off Control'
if this messes up, it i can cut fuel
does this help?
even better is this site, (has all the facts..)
http://honda-tech.com/showthread.php?t=810101

sounds more like the ECM than a relay, to me.

Are you sure it's running when cranking? It should have come on when key turned on.
This is for camrys up to year '91. It may apply in your '95, also. Check the 7.5 amp ignition fuse and the 15 amp EFI fuse and the EFI main relay, all located in the fuse box next to your battery. There is also the circuit opening relay tied in to fuel pump operation. It is usually found under center of dash. It is usually not the problem, but it is part of the checks for fuel pump operation.
If your car has a " check connector" or "service connector" (also called a fuel pump check connector) under hood on driver's side near the firewall (a little 1 inch cube with a cap on it), then bridge terminals +B and Fp with a jumper wire. Turn key to on (don't start car). The pump should come on. Listen for fuel return noise from the fuel pressure regulator, and verify pressure in hose from the fuel filter. If nothing, check those fuses above. Here is how power runs to pump: Battery to 40 amp main fusible link-to 15 amp EFI fuse-to EFI main relay-to circuit opening relay-to fuel pump-to ground. Ignition switch and the computer (ECU) work the relays to pass power to pump. Hope this helps some, Randy. Good luck.

This may just be a problem with the wiring of the car's ECU. It could be a loose wiring. The ECU may be receiving power intermittently. It could also be just an issue with the EFI relay. The ECU signal from the EFI is intermittent causing the flickering and the clicking sound.

Classic fuel pump failure, Efi fuse runs the fuel pump now its open and it won't start. With it cranking and not starting add a little fuel to the air inlet and if it (flashes off ) ( runs a few seconds ) there's no fuel and the fuel pump is the most common cause. The pump is in the tank and unles s you feel that you as good as any pro mechanic and have the experience then let a pro do the work as the danger is high even for a pro.

The codes are the following depending on your bike (EFI or Carb).:

P0373 CKP Sensor Intermittent Carb
P0373 CKP Sensor Intermittent EFI
P0374 CKP Sensor Not Detected Carb
P0374 CKP Sensor Synch Error EFI
U1255 Missing Message at Speedometer EFI
U1255 Serial Data Error/Missing Message EFI
U1255 Serial Data Error/Missing Message Instruments
U1255 Serial Data Error/Missing Message TSM/TSSM
this could be a number of potential things or maybe just a loose connection or ground somewhere. you are probably best to go to a shop, sorry.

Anyone with a Sprint, Daytona or S3 will tell you that this is normal. When you turn the key, the fuel system primes and the ECU does a self check for a second or 3. Wait for the self check to finish before trying to start the bike (so much for a fast getaway) On my RS, in the cold weather, the fuel pump is much louder than normal for a while, but once up to temperature, it goes away. Welcome to the electronic age.