The main aim of this paper is to propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, and to give the error estimations of approximate solutions about the electronic fast time scale component and the ion density deviation . In which, the bilinear element is used for spatial discretization, and a second order difference scheme is implemented for temporal discretization. Moreover, by use of the combination of the interpolation and Ritz projection technique, and the interpolated postprocessing approach, for weaker regularity requirements of the exact solution, the superclose and global superconvergence estimations of in norm are deduced. At the same time, the superconvergence of the auxiliary variable () in norm and optimal error estimation of in norm are derived. Meanwhile, we also discuss the extensions of our scheme to more general finite elements. Finally, the numerical experiments are provided to confirm the validity of the theoretical analysis.