It depends what the finish is. If its plastic woodgrain some black boot polish would probably do it.

If its real veneer then it depends...........

Hadn't thought of boot polish

What might Linn have used ?

I wondered about "touch up" even as I typed it !

I'll try felt pen in a not to visible place and see what happens

Not heard of blackboard paint
May be too matt

Felt pen has been applied on a small section.
I'll check how it is tonorrow

That's my next one then

Tried felt tip on an old Ikea table and it worked a treat

I got a chip out of the veneer on my KEFs (light wood finish) and bought some Ronseal touchup.
Complete waste of time and money - almost impossible to apply as per instructions and then the colour did not match the veneer, even though the touchup was the 'correct' colour.

Back to black car bumper renovator perhaps or ebony wood dye and then some clear wax or a spray and rub with pledge polish.
The shoe polish one is probably a good place to start.

I've transformed the look of loads of black ash kit over the years using a tip from a veteran box swapper:

He takes a can of either matt or satin black spray paint and an artists brush. Takes the top off the can and sprays a few puffs of paint into it. Using the small artists brush, he touches up any worn edges and little blemishes. Unlike marker, it doesn't show in the light and it dries in seconds. Done well, it's pretty much invisible. Matt is usually the better match for most black ash, although things like Proac have a higher shine and benefit from satin paint. I guess you could also repeat the process with a clear lacquer of your choice to seal the paint. Give it a try. I think you'll be pleasantly surprised.