Hi Graham,
As Keith says, this certainly sounds like a fault with the microUSB power connector. With a ‘positive’ connection, you should see a solid red light through the grille at the lower left side of the P3i Antenna end when power is applied - and this red LED should remain ‘on’ continuously until power is deliberately removed again. The fact that you say this light appears and disappears on gently moving the power cable supports the likelihood that this connection is the problem.
If it is this connection, the problem might of course be the cable end (easily sorted by replaceing the cable) though it is more likely to be that the socket on the main board itself has become distorted - which is generally best dealt with by swapping the Raspberry Pi board, - though I have had success by carefully reforming (gently tightening) the socket with a pair of fine pliers and then carefully applying ‘hot glue’ to support the joint once a tight connection has been re-established.
Keith is however presumably trying to determine depending on the age of your PAW whether this is a ‘warranty’ issue or whether the ‘upgrade’ route is the best answer in your case.
Regards
Peter