APOLOGIES IN ADVANCE, as I am a NOOB here.

I'm trying to sort out a Thermal Shutdown issue with a P3 that is only used for word processing and other simple tasks (reason it is not being replaced).
After checking things out with the BIOS toolkit and disabling thermal shutdown (for testing only), I monitored temps with both Speedfan and Everest (similar readings). What I noticed was that Temp1 (LM 78 chip) was all over the place, jumping from 42 to 120, then down to -110 c, with no discernible pattern to the jumps. The other two temps that Speedfan found remained nice and constant at expected values. The other thing I noted was that power monitoring (speedfan) also showed considerable fluctuation, with all 3 rails often showing well below target voltages.
I have checked and cleaned all cooling system parts and am happy that the cpu is NOT actually running too hot, but I would like to work out what is happening.

Could the fluctuations with the PSU be causing the LM78 to read incorrectly?

Noobs have to start somewhere - welcome!
I think this might be a problem with your power supply - and if you leave it the chances are that it will give up and maybe take something with it. If you can try another one and see.

In any event, your CPU cannot be running those temps. Something wrong with the sending circuit, sending you wild temps! It would shut down for sure at those figures. Your board has a problem with it's sensor(s). Don't worry about it. I would re seat your CPU to be sure it's as good as it can be. Use Arctic thermal 5 thermal grease or a high quality thermal grease.

Many thanks to both of you. I shall take ALL advice and see how things go.