OK,
Normally when a relay is operated it clicks. You can feel this if you put your finger on the relay. But, it only clicks once on change of state. It sounds as though your fuel pump relay is chattering when you apply power. Which makes me think the Motronic for some reason is pulsing the control to the relay.
During normal start operation when the ignition is first operated, the motronic sends a signal to the fuel pump relay for a few seconds to pressurise the fuel rail. The Motronic then waits for the signal from the HES to tell it the engine is rotating, when it receives that signal it reapplies control to the fuel pump relay and switch the pump back on. The engine should then continue to run.
I’m thinking the HES could be faulty sending a rouge signal to the Motronic. Which is then sending a spurious signal to the fuel pump relay.
There are a few things you could try, but I’m sceptical you will find anything.
Remove the fuel pump relay and make sure the 4 connectors on the fuse box housing are locked in place. Gently pressure should establish this. If the locking tab securing them is broken or the contact is not latched then as you push the relay home it will not engage correctly with the contacts.
Try removing Fuse 5 for a short while. This will allow the Motronic to reset.
Locate the HES connector and disconnect it. See if the fuel pump relay stops chattering when you go through the start sequence.
Ian