We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and the recovery of signals that are corrupted by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a recently developed uncertainty relation and provide corresponding recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge on the support sets of signal and noise.