Ok, I went to my local MG dealer today. A very helpful and knowledgeable engineer explained that it's not actually the socket, but the cable. The system is extremely fussy about cables (as many of you have pointed out in this thread). He produced the cable he uses for testing and plugged it into my phone and it worked immediately. He jiggled it about slightly and it remained connected. He unplugged and re-plugged a few times and it worked every time.
He's given me the part number and I'm going to order one. It's a VW cable and quite pricey. Part no. 000.051.446.BR)
So it's not sufficient to use an original Apple cable, it seems! (as I tried two of those).
Anyway, glad to be reassured that it's neither my phone nor my car that's at fault. A cable I reckon I can run to
Thanks again to everyone for your help and patience.