Sorry to read of your headlight issue, though in truth it is unlikely to be too difficult to resolve using a bit of logic. Yeah, modernday expectations of vehicle build quality and reliability are perhaps best not applied to 25 year old hand-built cars, sorry.
As for your fault it is almost certainly caused by a loose or dirty connection causing a RESISTANCE to the flow of electricity at that specific point, the resistance will convert the electricity to heat at that same point which will rob the light of volts when it is switched on. Thus the light will fail to illuminate as the voltage across it`s terminals will have dropped due to the aforementioned heat build up at the location of the bad connection.
I do not know the circuit on your particular Morgan, but some have a dim/dip arrangement as opposed to the standard high and low beam direct switching. If yours is the standard switching set-up then you are looking for that simple bad connection, and using the wiring colour code you can pick out the likely points in the circuit where a poor connection might exist.
Blue with a white tracer is the main beam circuit, and blue with a red tracer is the dip beam.
Any and every connection in the circuit has potential to be loose, dirty or just corroded, it is just a matter of finding out where that bad connection might exist, and as has been suggested the fuse holder or fuse it`s self has been found to cause issues in the past on other Morgans. I suspect there may be other connectors under the wings of the bullet type which may have suffered water ingress and subsequent corrosion...?
All connections need to be clean and tight and if left switched on when dirty/loose, the heat will build up will exacerbate the situation.
For wiring diagrams if you visit GoMoG.com, pretty sure you will find some diagrams and descriptions there, but beware that they may best be used as a guide as opposed to "as fitted" in your car, Morgans can vary to some degree even in the same model year and may also be modified by their owners over the years.
Good luck, you will find that folk here are happy to stick with you until you get it fixed.
