In this paper, we consider the existence of normalized positive solutions for the following nonhomogeneous Schrödinger-Poisson-Slater system \(\begin{aligned} \left\{ \begin{array}{l} - \Delta u + \lambda u + \left( {{{\left| x \right| }^{ - 1}}*{{\left| u \right| }^2}} \right) u = f\left( u \right) + h\left( x \right) ,~u>0,\quad in\;{\mathbb {R}^3},\\ \int _{{\mathbb {R}^3}} {{{\left| u \right| }^2}dx} = m, u \in H^1(\mathbb {R}^3),\\ \end{array} \right. \end{aligned}\) where the frequency \(\lambda \) is not fixed and instead appears as a Lagrange multiplier, \(m>0\) is prescribed, and h(x) is a perturbation. For the case where the nonlinearity f is mass subcritical and purely power-type, we establish the existence of a normalized positive ground state under the assumption that the perturbation h(x) meets certain mild conditions. Conversely, when the nonlinearity f is mass supercritical, we demonstrate the existence of a normalized mountain pass positive solution using variational methods, provided that h(x) is a radially symmetric positive function. This appears to be the first contribution addressing the normalized solutions for such a perturbed equation with a nonlocal term. Our findings may both generalize and enhance some of the recent results found in the literature.
