If it is a flash type circuit, the capacitor will be connected directly across the xenon tube, and a separate trigger circuit will generate a much higher voltag

If it is a flash type circuit, the capacitor will be connected directly across the xenon tube, and a separate trigger circuit will generate a much higher voltage to start the xenon conducting, and allow the capacitor to discharge it's energy and trigger the flash. So, it might be the 350v capacitor has failed and is not storing enough charge to generate a meaningful flash, or the trigger circuit which is not generating the initial >5kV to get the xenon conducting. Either way, i'd check the circuit before the xenon tube.