I went and had a look at lunch and you were, of course, correct: it only happens when the headlights are on.
I got hold of the previous owner (his telephone number is on some of the invoices) and had a chat - very friendly guy and didn't mind me calling at all. He had the washers removed because (and I quote) "they're f**king useless and covered the car in froth every time [he] used them".
I've got it booked in to have it looked at in a couple of days, I suspect it just means a pipe needs re-crimping.
Just out of interest, you mentioned that most people just take the wires off the pump, is there a separate pump for the headlights and is it easy to locate? Will disconnecting it cause any warning lights?