The spark gaps in projectors are used for igniting the lamp:
A capacitor gets charged until the threshold voltage is reached and the sparkgap fires, discharging the capacitor into the ignition transformer for generating many kV for igniting the HID lamp.
That's why it is called a switching spark gap (unlike other spark gaps, designed for overvoltage protection).
When the projector stopped working while it was on, I doubt the spark gap is the problem, because it is only necessary for starting the lamp.
Did you check for other faults?
I had a projector that didn't power up. After debugging the power supply without success I found the lamp door interlock switch broken off its mounting bracket, therefore it disabled the lamp power supply.