In this paper we study the nonlinear Schrodinger-Poisson system with a perturbation: \begin{equation*} \begin{cases} -\Delta u+u+K( x) \phi u=\vert u\vert ^{p-2}u+\lambda f(x)\vert u\vert ^{q-2}u \text{in }\mathbb{R}^{3}, -\Delta \phi =K( x) u^{2} \text{in }\mathbb{R}^{3}, \end{cases} \end{equation*}% where K and f are nonnegative functions, 2≥q≤p≤6 and p≥4, and the parameter λ∈R. Under some suitable assumptions on K and f, the criteria of existence and multiplicity of positive solutions are established by means of the Lusternik-Schnirelmann category and minimax method