Previous studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non-convergent. In this work, we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature. For finite elements of arbitrary order, we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator. We compare the proposed approach with the pseudo-derivative method developed by Belytschko and co-workers and, for linear elliptic problems, with our previous remedy that involves splitting of polynomial and non-polynomial of elemental energy bilinear form. We present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals.