In this paper, we consider a class of elliptic problems in a periodically perforated domain with $L^1$ data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to $H^1$, the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself.