So no primer, just electric pump in tank.
Just a thought, when you go to start it as usual when fault occurs, try turning ignition on, listen to the electric pump until it stops, do it a couple more times, then see if it starts without an issue, as that may chase any air out of the inlet side back to the return to the tank.
Also when you do this, see if you hear any air bubbles in system.
One of my daughters had a 1.3MJ in a 06 Fiat Grande Punto, it never cut out, but from cold it certainly wasn't a race horse.

I have a 2014 Vauxhall Combo 1.3MJ, I think that is the later engine, but still pretty poor from cold, it quietens down as it warms up, but the 1.6 is a better engine in my other Doblo and no rubbish timing chain issues, although neither have any old school diesel grunt!