We present a modified Poisson-Nernst-Planck (mPNP) model to include both non-electrostatic effects and electrostatic correlations on ion dynamics near charged parallel blocking electrodes. The first contribution appears directly on ion chemical potential, whereas the latter is addressed by a modified Poisson equation. The model consists in a partial-differential-algebraic equation system, solved by the DASSLC code, implemented in EMSO simulator. Monte Carlo simulation results were properly described without the aid of fitting parameters. The model predicts overscreening for 2:1 electrolytes, which was not accessed by classical PNP-like approaches. Alternate voltage simulations show a meaningful effect of electrostatic correlations on mobile charge dynamics for 2:1 electrolytes, where ion-ion correlations are stronger near the electrodes. This is the first report accounting for electrostatic correlations on ion dynamics via PNP-like models under ac voltages to the best of our knowledge.