We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on $L^p$ spaces on bounded domains of $\mathbb{R}^n$ with a nonlinear drift term given by the superposition operator generated by a monotone function on the real line with power-like growth. The noise is of additive type with respect to a cylindrical Wiener process, with diffusion coefficient not necessarily of $γ$-Radonifying type.