We build a class of stochastic gradient integrators with second order accuracy via noise correction. The integrators multiply the injected Gaussian noise of the stochastic gradient Leimkuhler--Matthews integrator by a noise factor, obtained by applying a scalar function, the noise corrector, to the eigenvalues of an estimate of the covariance of the batch gradient from the same batch. For nonconvex potentials that are strongly convex outside a ball, every noise corrector satisfying a polynomial bound and every batch size, we prove that the numerical bias is of second order in the step size $h$ and that the mean square error of the time average of $N$ samples is bounded by a constant times $(Nh)^{-1} + h^4$. Under Gaussian gradient noise and an additional polynomial bound on the square of the noise corrector, we separate the coefficient of $h^2$ in the numerical bias into explicit terms from discretization and noise correction.