We report results of a search for dark-matter-nucleon interactions via a dark mediator using optimized low-energy data from the PandaX-4T liquid xenon experiment. With the ionization-signal-only data and utilizing the Migdal effect, we set the most stringent limits on the cross section for dark matter masses ranging from 30 MeV/c^{2} to 2 GeV/c^{2}. Under the assumption that the dark mediator is a dark photon that decays into scalar dark matter pairs in the early Universe, we rule out significant parameter space of such thermal relic dark-matter model.