Lots of questions tonight...

The old cooker hood in my kitchen has a 4" hole at the right height, and
about as far left as I can possibly have the new hole if I'm having a
chimney hood.

Modern cooker hoods appears to want 5", or more usually, 6" holes.

Is it possible, using a core drill, to drill over the the top of an
existing hole? Or is it going to wander so far that I end up with a
complete mess? Obviously I'd need to offset the centre at the same time,
so that the left edge of the hole didn't move any further left.

Never used a core drill, no idea if this is going to work or not. The
alternative is to find a hood that takes a 5" hole and use the 5"-4"
reducer I have hanging around in the garage.

Ben