Clarity, applied onto background with a local adjustment brush or gradient, is quite good for this purpose... when set to a negative value (Negative and Positive Clarity are very different in what they do). Combine in whatever else you want, e.g. a change of tonal contrast, to this same adjustment. Also negative DeHaze looks somewhat like fog, again working very differently than when the slider's set to positive values. Whichever adjustments you apply over the general background, you can then subtract your subject out from those. Conventionally with an erase brush, or if you have the new v11, maybe with the help of "select subject". Your subject will then be left free of the background "fog", showing only the effects of your global adjustments, as modified by any other local adjustments you may employ too.