You correct any light that is not "white", that needs to be, if the light is used as a Key light - I.E. a primary source for the camera. You can go nuts trying to correct all the assorted higher kelvin lamps that suddenly become Key lights as a result of the camera(s) moving around a lot - side shots, back shots looking towards the audience,
etc...
This is where a TV/Film lighting director/gaffer - DP earns their pay, figuring out where the camera shots are going to be, then designing Key and Fill as needed for all the shots. It's a tough balance, as you may want the arc source flash & trash ML's to keep their "whiter" look, as it's punchier and flashier. Problem comes in when one of those "Effect" units suddenly is focused on a performer with a camera in the same angle. The "Effect" unit, without color correction (to keep it punchy and flashy) is now in the wrong color to be a Key light. That can be annoying if seen repeatedly, but sometimes you can get away with it if it's random.
One thing that's useful is to determine in advance, where the camera(s) are going to be, if there's more then one. Then plan on Key lights from incandescent sources to cover those angles and not rely on arc lamp ML's as keys, but to keep them off the performers if possible.
SB