I was just wondering, is the capacitor on a CWA ballast required for proper operation, or is it just to increase effenchy? I am reffering to the type for probe start MH with a capacitor connected between the CAP lead on the ballast and the center contact on the lamp socket.

Yes, with CWA, the capacitor offers both power factor correction and adds to the ballasting.

I believe even with most multi tap HX, you're supposed to use the provided capacitor as the primary coils are designed for a current draw with the capacitor in mind.

Single tap HX and reactor ballasts don't usually need a capacitor.

I was just wondering, is the capacitor on a CWA ballast required for proper operation, or is it just to increase effenchy? I am reffering to the type for probe start MH with a capacitor connected between the CAP lead on the ballast and the center contact on the lamp socket.

The "magnetic" part is an autotransformer, with a saturable magnetic shunt between primary and secondary, the shunt designed so it act as a saturable (current dependent; current above saturation limit causes the inductance to decrease) series inductance seen from the output, so the output acts as a 220..300V voltage source in series with a saturable inductor.
This inductance forms, together with the series capacitor, a series LC, with its resonance tuned above the 60Hz mains, so the capacitance reactance is bigger.
The total reactance in series with the lamp is then Xc-Xl, as they have opposite phase. That means one interesting feature: Up to some current, where the inductance is large, the difference is small, so the current is not that much limited. But as the current increases (e.g. due to rising mains voltage,...) beyond the designed saturation point, the inductance starts to drop as the shunt saturates. But lower inductance means the overall ballasting reactance (Xc-Xl) actually increases, so counteracting the current increase. In this way the current is maintained pretty tight around the designed level, even over quite a wide mains voltage variation, keeping the lamp power constant, hence the CanstantWattage in its name (the Wattage is actually not correct, it came from time when the only variation source was the mains, not the lamp).
The overall circuit is then of a capacitive phase shift nature towards the supply. In order to bring it back to resistive phase, a compensation inductor is needed. And that comes in the form of an air gap in the primary part of the magnetic circuit.
So the capacitor (along with the saturable magnetic shunt between the windings) is there the key ballasting component, but the power factor correction is the primary gap in the "magnetic" part.

Thank you!

CWA requires the capacitor for proper lamp operation, and HX is the same as series choke, and don't requires a capacitor.