@OP I'm most interested in what you told the police.
The question is how
the car was being
used. If the car had music blaring, it's being used as a beat box and depending on his taste in music, Plod might reasonably believe that it's "likely to cause annoyance" -
PRA 59(1)(b). But note that it must be
being used now in a manner likely to cause annoyance. Not
is likely to be used in such a manner.
If the uninsured twoc'd the car and drove it, then it looks to me like
the car was being used in a manner that contravenes
s. 34 RTA - "driving without lawful authority" on "land not forming part of a road", and so Plod reasonably believes
the car is being used in a manner etc engaging PRA 59(1)(a).
Tbh, if I was using the car park and could see numpties not in control of their vehicles, I'd be annoyed at least, and possibly alarmed, - so the "learn and grow up" advice seems well placed to me.
Section 59 (1) of the Police Reform Act 2002 establishes that where a police constable in uniform has reasonable grounds for believing that a motor vehicle is being used on any occasion in a manner which contravenes section 3 or section 34 of the Road Traffic Act 1988 (careless and inconsiderate driving and prohibition
You missed a bit:
... of off-road driving)
(2)A constable in uniform shall also have the powers set out in subsection (3) where he has reasonable grounds for believing that a motor vehicle has been used on any occasion in a manner falling within subsection (1).
If police turn up and it all looks a bit chaotic and they reasonably think they missed the main event but it wasn't pretty, looks like they can issue the warning.
I'm with
@Lucan Unlordly on waiting for the paperwork though: that will give the copper's reasons, which you could in theory challenge. That would either be by (1) persuading his super that the PC could and should have applied his discretion in a different way (even though he had reasonable grounds to issue the warning), or (2) by persuading the super or a court that no police officer could have reasonably believed what the PC wrote on the docket. Pretty tough, given the latitude PRA gives them - classic Blair/Blunkett legislation, that, and things don't look like they're improving.
But if he wrote "seen on CCTV doing trolley stunts near car" I reckon it's on the border of irrationality (at least) to go from there to believing that
the car was being used etc...