General: Sony MZ-RH1

Lolly, you seem to have the same PSU, so what is causing your hum is a mystery to me. As I said, no ferrite cores have been fitted to any of the cables, except the one moulded on the PSU cable.

On page 37, the manual actually tells you to plug in the PSU before recording from an external source via the optical/analogue input. I assumed this was a battery limitation thing as the current drawn when recording is higher than when playing.
I wonder what is causing your hum? If it works on battery OK for you, I guess it doesn't matter. Strange none the less.

I've never had the issue when recording from line in or optical, only from the microphone. It could possibly be the microphone, actually.

I have a very good microphone! But it's not shielded. Normally it's OK because it's suspended in mid-air from an organ loft somewhere!

Organ lofts, surprisingly enough, seem to have plenty of power sockets - probably on account of the organ's own electronics and lights - which are often supplemented by angle poise lamps for more light, which need to be powered from somewhere...