  In this paper we study the convergence of integral functionals with
$q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$.
Under the assumption that the perforations are small balls whose centres and
radii are generated by a \emph{stationary short-range marked point process}, we
obtain in the critical-scaling limit an averaged analogue of the nonlinear
capacitary term obtained by Ansini and Braides in the deterministic periodic
case \cite{Ansini-Braides}. In analogy to the random setting introduced by
Giunti, H\"ofer, and Vel\'azquez \cite{Giunti-Hofer-Velasquez} to study the
Poisson equation, we only require that the random radii have finite
$(n-q)$-moment. This assumption on the one hand ensures that the expectation of
the nonlinear $q$-capacity of the spherical holes is finite, and hence that the
limit problem is well defined. On the other hand, it does not exclude the
presence of balls with large radii, that can cluster up. We show however that
the critical rescaling of the perforations is sufficient to ensure that no
percolating-like structures appear in the limit.
