No, I don't think that's it.
I think he really was sunburned from actually doing his job for once.
Well she did make a negative implication about his appearance to which he retorted weakly something like "Maybe it was from doing his duty (checking pawn shops?) out in the sun". So, kn51's take makes sense and does not preclude that you are also correct in her assuming wrongly.
This season they seem to be making a similar effort to avoid the dreaded procedural but it's not really clear where we're going - as mentioned by a few others previously. Are we trying to solve Caspere's murder? I don't get the feeling that anyone really cares about that. Are we more concerned with where the state investigation is going? That seems to be the main focus since it's what brings the cast together.
Yeah, S1 began as a murder investigation of a sympathetic victim that turned into a wider unresolved conspiracy (and the perpetrator was presented quite obviously early on). Whereas with S2, the only reason to care about the unsympathetic victim Caspere (or any of these ne'er-do-wells) is their part in the corruption conspiracy from the start (and now how entangled).