Well, I took the turntable into my local Practical Hi-Fi. They changed the cartridge, largely because the same one had been on for some time and it was due for a replacement and declared the turntable fine after listening to it for some time.
I took the turntable home and, within five minutes, the hum (at a very low frequency) had returned. This hum only occurred with the turntable's power connected to the mains (previously it hummed whether the power was connected or not). So, I took both the turntable and amp to Practical Hi-Fi for further investigation.
This fault being somewhat erratic, it took the best part of 15 minutes to manifest itself in the shop. With the turntable connected to another amp it behaved perfectly. So, we plugged another turntable into my amp, again its behaviour was perfect. I've now left both with Practical Hi-Fi for further investigation in the hope that the fault can be traced.
This is basically to thank Practical Hi-Fi for their excellent and patient customer service on a pair of items that are each at least eight years old.