We study the convergence of integral functionals with q-growth in an n-dimensional bounded domain, with n>q>1, which is perforated by a random number of small spherical holes with random radii and centres. Assuming that the latter are generated by a stationary short-range marked point process, we show that in the small-perforations limit we obtain an averaged nonlinear analogue of the capacitary term obtained by Cioranescu and Murat in the linear deterministic periodic case. We only require that the random radii have finite (n-q)-moment, which is the minimal assumption to ensure that the expectation of the nonlinear q-capacity of the spherical holes is finite. Although under this assumption there are holes which overlap with probability one, we show that the clustering holes do not have any impact on the homogenisation procedure.