running lean means the engine is not getting enough fuel in the mix

i found some pretty good info

air/fuel


lean or rich they both suck but i'd take rich before lean

Is a lean mix enough to cause overheating? (I am having an overheating problem and a BHG does not seem to be the source. Used a block test kit and found no exhaust gas in hte coolant. Looking for other things to try...)

thats what the book says . and it is a fact .
are you throwing codes ?
are you bubbling ?
is this a turbo

no exhaust gas in the coolant is a great sign. Yes, you can get hot from being lean. If one of the previous owners thoughts it was wise to run only water in the radiator, you're radiator is gonna clog and not be able to cool worth a flip. If a new STOCK radiator doesn't fix it, check fan clutch, shroud, fan, thermastat, radiator cap, and waterpump.

If you're not bubbling, and the gauge just looks hot, it might be the gauge. Also, you could just be a bit paranoid, let it idle and heat up fully, whatever is shown is normal.