Last night I put a cleaning CD in it and - behold it started to play
it. However, on some CDs it still doesn't recognise them and play them
- just the clicking (lens hitting the CD as it tries to focus??).

The LASER LED is nearly dead. The clicking is the focus server hitting it's
endstops trying to 'see' something recognisable.

It is possible to replace the LASER, but usually bloody tricky. I've
generally given up after an hour or so of each that I've tried to replace.

I think the claening disc is a red herring. I've never, ever, known one do
any good whatsoever. Nice quick profit for someone though (cheap disk,
little brush, looks like magic, doesn't work).

Al.