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Is mixed lighting better off corrected or let be depending on the photo?
That is a REALLY SUBJECTIVE ... depends.
For a LOT of people, they are fine with the mixed colors ... going along with fact that it is a property of light & color. This is part of the reason we are able to get the really cool colors at different times/places in nature. But it can also be part of the reason we get strong blue shadows, blue hair, cyan wedding dresses, blue snow, etc. Some folks just roll with it, others don't. Some people think color correction looks unnatural, others think it looks correct.
Me, I lean pretty strong toward correction to at least close to neutral, then if I want to have it "toned" for mood, that's easy enough. Mostly I strive to find my dominant/prominent neutrals and get them close to neutral. In the case of your stairs, stone work ... one person might find the warm/cool juxtaposition really cool and intriguing, while another says the color is off.
Like I said ... really subjective @ depends.
As to the correction, I tried to isolate the blue channel and warm it up on one layer. Then I tried to isolate the red channel on another layer and cool it down. As you can see, this left the greens unattended to a bit and wasn't a great job because I've not got some blue in some areas that probably are neutral.
Desaturation and masks can be an effective strategy also. I tend to strive for color balancing neutrals first because while desaturation can reduce the offending areas in the neutrals, it doesn't really make an adjustment for the cast in other non-neutral areas as much. Imo, reducing casts, improves clarity in addition to making the colors more accurate, so it is my first choice when I can see to do it that way.
As to the "how" ... mostly I check a scene to find either areas that I know I want to be neutral and the simply tweak on color balance sliders until I equalize the R,G,B numbers, or if I've got a color that is NOT neutral (say a blue shirt) but I know it is being lit by two different light sources and compare the two sets of RGB values (which would be the same if it were the same light). Reading the differences between the two sets will show me have much difference there is between the two light sources. Then, I have a decision to make at whether I want to try to "split the difference" or allow one to be "correct" and then adjust the other one closer to it.
The key to most of it is to be able to recognize it the lighting is predominantly from the same source or from different sources. Most people associate outdoor lighting with just one color of "daylight" but you stairs example is a great one to illustrate that outdoor lighting can actually be multiple/mixed. If you've got only one primary light source, then global correction can work pretty well. But if you truly have very mixed sources, then selective correction is likely more warranted. At which point most people just "go with it".
As to your potato chips ... there are different things that you can do that I refer to as S&P to taste. regarding how much contrast, sharpening, saturation you want to put into your image. Stick around and you'll readily realize we can often times have a wide variance of opinion on measure and approach ... as others show their input as well.