We report on electric-field-tuning experiments of single chromophores in a transparent matrix on top of a microcrystalline layer of silicon carbide (SiC). Upon changes of the electric field strength, most single-molecule lines show a slow creeping behavior toward a new equilibrium position. We attribute the relaxations to slow charge rearrangements in the SiC on a (sub) microscopic scale. This interpretation is supported by atomic force and scanning electron microscopy images of the SiC surface, which show mosaic structures with a typical grain size of about 300 nm.