We present a new covering theorem for a nonlinear mapping on a convex cone, under the assumptions weaker than the
classical Robinson’s regularity condition. When the latter is violated, one cannot expect to cover the entire neighborhood
of zero in the image space. Nevertheless, our covering theorem gives rise to natural conditions guaranteeing stability of a
solution of a cone-constrained equation subject to wide classes of perturbations, and allowing for nonisolated solutions,
and for systems with the same number of equations and variables. These features make these results applicable to
various classes of variational problems, like nonlinear complementarity problems. We also consider the related stability
issues for generalized Nash equilibrium problems.