We prove by means of the Moreau-Yosida regularization an existence result for perturbed degenerate state-dependent sweeping processes with convex and nonconvex moving sets. The moving sets vary in a Lipschitz continuous way with respect to the truncated Hausdorff distance. The theoretical results are applied to obtain the well-posedness for the online mirror descent method. The novelty of this work lies in the introduction of new local arguments for the Moreau-Yosida regularization that allow us to obtain the existence of solutions in small intervals and then pass to any interval.