We introduce a kind of radially polarized partially coherent (RPPC) beam with a prescribed non-uniform correlation function, called a radially polarized cosine non-uniformly correlated (RPCNUC) beam. Based on the extended Huygens-Fresnel principle, we study the propagation properties in free space and in a turbulent atmosphere. Unlike RPPC beams with uniform coherence, RPCNUC beams possess the invariance of dark hollow cores and radial polarization, and exhibit self-focusing properties. In a turbulent atmosphere, the intensity distribution demonstrates self-healing properties over a certain propagation distance. We also investigate how to adjust the beam parameters to reduce the turbulence-induced degradation in detail.