The auxiliary pump is the one mounted on the radiator inside the engine bay. It’s there to keep coolant running through the turbo after shut down. Access to the plug is pretty easy. Just meter the motor terminals and see if it’s open circuit. You could even put 12v on it pretty easily to hear it run. If the motors ok it’s probably just a dud relay.
I agree with gav. If the car has been sitting then there’s every chance the vvt cam chain (different to cam belt) tensioner either lost its prime or had crusty old oil gunk in it. I’d just change the oil and reseat the solenoids plug a few times after a bit of a spray with crc and monitor. It’s only used by the engine during first cold starts along with the sai for emissions so that’s the only time you’ll get that error.