Updated: Oct 22
In photography the adjusting of colors to match reality is called White Balancing based on the original technique of using a pure white or pure gray source to establish real colors in developing software such as Photoshop and Lightroom.
How do you create the correct white balance for 2 competing light sources, like this scene from the 2019 Tax Executives Institute 75th Anniversary meeting?
I do it by separating the elements and adjusting their balance independently before saving the composite as a JPEG. You can also save the layers as a .psd or .tiff file if you plan to do more work on them.
In this case separating the image on the screen behind the ladies wasn't difficult in Photoshop. For more complicated images the selection tools available in Photoshop will make most selections pretty straight forward even if consuming more time.
In both the Photoshop Camera RAW filter and the developing tool I use most, Lightroom, also from Adobe, the white balance tools change the "color temperature" of the image. That's the first thing I'll try but other tools exist in both applications that will allow fine tuning of the element in question so you can match the look of the main image quite well.