The pump is slow to start. It cranks for a few seconds before running at full speed. sometimes, after cranking for a few seconds, it shuts off. The unit is 10 years old.

When you put your key in, don't turn it all the way but to the point where everything powers up. You should hear a hum from the lower rear that lasts about 3 seconds. That's the fuel pump. If you don't hear that, it could be the fuel pump.

indicates air in the pump
very slow leak to allow pump pressure to drop to a point where it starts but then the supply is below requirement until the continued cranking removes the air and then the engine starts and runs

You do not say what vehicle you have but a possibility is that the computer is not getting a signal that the engine is cranking over and when that happens it will shut the fuel pump off in a few seconds. It does the same if you just turn on the ignition and do not try to start the car. The pumps runs for a few seconds and is then shut down by the computer because there is no engine crank input..

The way the newer systems work the engine spins at least a few turns while the computer checks everything for correct readings. In the old days it was "cool" to be able to reach in and "bump" the key and have the car start. It meant you had the car tuned perfectly. Now the vehicle computer will not turn fuel on until it is happy with readings. One trick you can try is turn key to full on position and let it sit for a second or two before turning key the rest of the way to crank. This allows the system a second to do it's checking without wasting power cranking. Had Nissan that was just like this. Just wait a few beats before turn to full crank. You can also listen for fuel pump whine in that few seconds, and then hopefully whine to stop when pressure hits full. If whine never stops fuel pump may be getting weak. If you can get access to higher end code reader you can also check for "pending" codes that may be showing up sometimes (holding start up) but then working (so engine finally fires).

first of all, check the fuel pump relay to see if it is going bad. if it is ok! also try this, Have a person turn key to on,, and put your hand on bottom of fuel tank. if the pump is turning on, you will feel a vibration in tank. you may have to do this a few times. the pump only turns on for a few seconds to pressurize system, or to get fuel to the engine. just keep turning key on and off to recycle fuel pump. Good-Day! Hope this helps! relay for pump may be bad.< Check this first.

you failed to say, does crank, when it don't start?
and btw, the fuel pump on most cars only run cranking or running
some run for 3 seconds, key on, but you'd miss that fact. using
only one person.
the key word here is crank, does it crank as full speed, via the cranking starter motor. that is the engine cranks spins fast.
on this car it will spin min 250 RPM, cranking, and uses large
battery current , so on diesels it takes a very strong battery to start
and working glow plugs.

That is a possible cause alright. I would get a can of fast freeze spray from a motor parts factor. If its ok when cold but shuts off when hot its very likely.when the engine shuts off, spray the crank sensor and immediately start engine. If it starts and stays running its your crank sensor. Does it have a duel mass flywheel?? Some times if the crank sensor picks up off the flywheel and the duel mass is worn , it then runs out of sync with the cam shaft pick up sensor, which will cut the vehicle off. Just a shot in the dark? If you can drive vehicle when cold , check for vibrations whilst engine is cruising at steady speed. You will not feel vibrations under load. Keep us informed on how you get on.

have you checked the pump relay sounds like your relay is ready for a change

the TPS senser is bad its on the injecter pump or you have a bad crank shaft sonser

TRY THIS BEFORE U SPEND ANYMORE MONEY N PARTS.
Here is the common cause of surges, stalls at stops, slow idle speed, erratic idle speed, rough idle and engine hesitation (and other problems), it is in most cases the idle speed control air-bypass valve and or throttle valve and upper intake, these area's get full of gunk and combustion residue over the miles and cause idle issues (stalls, low idle) like yours, Get a can of intake cleaner from any local parts store, not carb spray, intake cleaner, it is made by a company called CRC, remove the air intake hose to the engine, hold the idle high so the engine won't stall, then spray the can of cleaner into the intake while keeping the engine running, use at least 1/2 the can, shut down the engine and disconnect the battery for 5 minutes, then restart and complete a number of mixed driving cycles, town, freeway, stop and go etc., after a few days the problem will go away as the system will relearn to the clean intake