Work is getting in the way of fixing my toys . . .
"the pump runs when I apply 12 Volts" Can you explain how you did that.
If you connect a battery to fuel relay pins 85 and 86 you should hear a click and then have continuity between 30 and both 87 pins.
A very easy way to test a relay is to put a 9V battery to the 85 and 86 terminals and listen for a click.
I have performed all of the tests suggested on this site and so far - no joy.
I tried it again this morning hoping the boogey man came by and helped me out - no such luck.
It's starting to sound like something is supposed to happen when the ignition is turned on and the starter pressed but isn't. Is this where the Hall effect unit would /should be kicking in?
With the jumper wire between 30 and 87, was the pump running?
One problem that has arisen is that you think you're working on a toy.
The next problem is that questions are being asked that you don't seem to be answering. Additionally, you assert you have power upstream of the four-pin fuel pump connector. How have you determined whether there is power downstream of the connector? Also, The boogey man causes trouble; helping is not in his job description. Finally, you might have performed all the suggested tests but you don't seem to have told us all the results.
To answer your Hall sensor question:
The fuel pump relay is activated during starting and it stays activated until the signal from the Hall sensors drops out, e.g., when the starter is shut down, the engine stalls, or the running engine is intentionally shut down.