  We study the distribution of the maximum of a large class of Gaussian fields
indexed by a box $V_N\subset Z^d$ and possessing logarithmic correlations up to
local defects that are sufficiently rare. Under appropriate assumptions that
generalize those in Ding, Roy and Zeitouni (Annals Probab. (45) 2017,
3886-3928), we show that asymptotically, the centered maximum of the field has
a randomly-shifted Gumbel distribution. We prove that the two dimensional
Gaussian free field on a super-critical bond percolation cluster with $p$ close
enough to $1$, as well as the Gaussian free field in i.i.d. bounded
conductances, fall under the assumptions of our general theorem.
