  We investigate the effect of correlated disorder on Majorana zero modes
(MZMs) bound to magnetic vortices in two-dimensional topological
superconductors. By starting from a lattice model of interacting fermions with
a $p_x \pm i p_y$ superconducting ground state in the disorder-free limit, we
use perturbation theory to describe the enhancement of the Majorana
localization length at weak disorder and a self-consistent numerical solution
to understand the breakdown of the MZMs at strong disorder. We find that
correlated disorder has a much stronger effect on the MZMs than uncorrelated
disorder and that it is most detrimental if the disorder correlation length
$\ell$ is on the same order as the superconducting coherence length $\xi$. In
contrast, MZMs can survive stronger disorder for $\ell \ll \xi$ as random
variations cancel each other within the length scale of $\xi$, while an MZM may
survive up to very strong disorder for $\ell \gg \xi$ if it is located in a
favorable domain of the given disorder realization.
