I was going to post a question about this asking for help, but I've since solved it. As it's been doing my head in for weeks, if not months, I thought I'd detail what's been happening and the solution here.

The machine won't restart or start if it's been shutdown recently (within half an hour or so). I press the power button and the power light flashes dimly, once a second and the case fans sort of judder, they make about a 10th of a revolution and then settle back to where they started. It won't even POST.

Pressing the power button again, or even holding it for a long press immediate shut down does nothing. Same with reset button. The only way out is to pull the power cord and leave it for at least ten minutes.

I've fiddled with the BIOS and internet searches didn't really give any clues until after I'd sussed out what it was.

So, the cause? A powered USB hub, specifically this one, although I didn't get it from the retailer mentioned in the link.

Disconnect that and all is fine. In fact, even if the machine is in it's fan almost spinning state like I described above, unplugging the hub will suddenly allow POST and boot.