Also,using the PF cap can make your ballast run cooler rather than using power to heat the ballast.Usually there is a cap value marked on the ballast nameplate tells you the cap value and voltage.I added one to a mercury ballast-250W and it runs much cooler.

When speaking about justr PF correction caps (so not the series connected ballasting caps in CWA or resonant caps in mag-reg)
The ONLY case where the PFC cap may really influence the ballast losses (so temperature) is with multi tap HX transformer ballasts, where the cap is at higher voltage tap than the supply connection (there the reactive power then uses bigger part of the primary winding).
But if the capacitor is connected straight parallel to the mains, there is no influence on the ballast temperature possible whatsoever.
And if the cap is connected to a lower voltage tap than the mains input (e.g. 120/208/244V ballast has the capacitor connected to the 208V tap, but it is supplied from the 244V), the capacitor connection actually increases the ballast losses, so it's temperature (because here without the cap the reactive power comes only from bigger part of the primary winding). But that is really rare - just exactly for that reason the PFC capacitors are usually specified to be connected across practically the complete primary winding.

I have the included capacitor installed in parallel with the line in feed before the ballast. If I remember correctly, this lamp and ballast combo would measure .51 amps between the bulb and ballast. Based off what I've read on here on other posts, this current would indicate that it's being slightly under driven by about .1 amps. I've never measure the arc tube voltage though.

I have the included capacitor installed in parallel with the line in feed before the ballast. If I remember correctly, this lamp and ballast combo would measure .51 amps between the bulb and ballast. Based off what I've read on here on other posts, this current would indicate that it's being slightly under driven by about .1 amps. I've never measure the arc tube voltage though.

Indeed, the .51A is about 15% below what the lamp should ideally get, but it is not that bad.
The arc voltage is dictated by the lamp and with MV that will be really close to the rated 90V or so (when fed with sinewave current, the lamp has power factor of 0.9 due to shape mismatch - voltage is near square wave, current near sinewave), quite independent on the actual lamp current.