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Hi and welcome.
Looking at the pic I'd say what you think is a burn mark is actually barium, its put there to absorb any air left in the envelope after manufacture.
Is that all of the circuit?, vfd's usually run at 30 to 40v so I'd expect a little inverter with a transformer or choke.
I'd be looking at the blue cap and the transistor to start with, then maybe check the vfd heater (very fine wires running across the front of the display, but its unusual for the whole thing to fail).
I've had 2 metro's, didnt mind them, but I didnt like the way the wings rusted behind the headlight bowls.
The "burn mark" is what is called a Barium getter.
To improve the vacuum beyond what can by achieved by mechanical means, a small barium disk is placed inside the sealed glass envelope.
Via induction heating, the barium is sputtered and it deposits on the inside surface of the glass envelope. The getter is located in a way that won't obstruct the visible area.
When new, the barium deposit is a silvery, very shiny color.
Yours looks brownish, clear sign that the barium has been reduced by gas. This is usually the result of a leak which slowly allows the atmosphere to seep back in.
When that happens, the cathode material in the filaments (the three parallel white lines running from left to right in your first photo, right in front of the digits) degrade and stop emitting electrons.
Is that a vacuum tube fluorescent display? I love those displays.
Sorry to hear yours isnt working as well as it did when new.
The only fix is probably to either replace the display (which wont be easy) or just replace the whole clock.
They have these clocks, or at least had these clocks, on some sort of surplus electronic parts sites. You could check around the web for a full replacement. That would be quick and easy.
My car has the carpy LCD type display. I hate those things
The heater is usually the outer 2 connections on the display and will read 3 ohms or so, but you can if get a magnifying glass you can see the heater wires, they are nearest the glass above the numbers, when they fail the wire breaks, unless the connection goes bad.
Is there a choke or tranformer of any kind, its driver transistor is also a likely fail.
What does the display do, nothing?, or something weird.
How is the clock set, are there buttons built in, or are they somewhere else in the car?