jamesmartin said:

yours must be edc then ,edc fuel pumps do this with ignition on ,glow plugs shouldnt make a real difference on tdi unless exceptionally cold , poor starting is usually down to low compression easily checked as will breathe heavy out of filler if cap removed but not so likely on tdi, or fuel issues ,like air in fuel,retricted fuel .worn pump ,poor injectors most of which would give other symptoms when running,air would be place to look first

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The car is worth about £1500. As it is there just to tow the caravan and use for work on really bad days (when I can't use the bike), I think I might just leave it.

A big thank you to all those who have posted replies here.

The car isn't worth spending days and loads of money on. I have ruled out the obvious with your help. I might just let it develop.

Cheers, John.

checking good clean fuel is cheap ,and worth doing clear plastic pipe gives proof ,if there is air, pump can get damaged eventually and it will eventually let you down

EDC equipped 300TDI's need their glow plugs to work on cold mornings,unlike the mechanically governed pumps.They have a much lower start quantity.
Poor starting from cold with EDC cars is usually failed glow plugs,poor earthing,perished/loose leak off pipes,rusty fuel pick up pipe or a failed coolant temp sensor.It may be worth getting it on Testbook to see what the EDC ecu is seeing.Its a good,reliable system much maligned by those who dont understand it.