It seems that the limit must exist almost everywhere and if $\mu$ has no atoms then it coincides with $|averaged \ lim \ p_n|^2$. For point masses some extra summands appear. If so, I am sure this must be known.

Yes, and another simple example is the Dirac measure at a point of the circle: then the averaged limit exists, say, for the Cesaro summation method. There are some reasons to think that the fact can be true for every complex measure and that it can be already known. I would very much appreciate any references/arguments.