In this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like fuγ and a lower order term bu, where u is the solution and b and f are only assumed to be summable functions. We show that, despite the lack of regularity of the data, for suitable choices of the function b in the lower order term, a strong regularizing effect appears. In particular we exhibit the existence of bounded solutions. Worth notice is that this result fails if b≡0, i.e., in absence of the lower order term. Moreover, we show that, if the singularity is “not too large” (i.e., γ≤1), such a regular solution is also unique.