Well, it sort of works.
I cobbled this together last night from a small IKEA desk light as a "proof of concept".
This one is another 20 cm longer than another suggested earlier.
My thanks though for highlighting that one as I probably wouldn't have found this one without that heads-up.
I used a thick plank of wood behind the headboard as a "counterweight" with two more bits of batten to form the "hook". The desk lamp comes in two pieces anyway so no disassembly was needed.
It nicely fills the gap in the light cast by our bedside lights and we can now do crosswords without eyestrain. I did think about adapting some sort of clamp but this just lifts off and the weight of the plank holds it perfectly steady. I would have preferred a switch on the light but it ticks all the other boxes now.
Of course I should pretty it up a bit but I won't be surprised if that takes another 10 years or so. ;-)
Tim