We explore the propagation characteristics of a partially-coherent radially-polarized vortex beam (PCRPVB) with Gaussian Schell-model correlation structure and, using the extended Huygens-Fresnel principle, derive analytical formulas for the average intensity of PCRPVB propagating through the non-Kolmogorov atmospheric turbulence along a slant path. Numerical results show that the normalized initial profile with a hollow distribution of PCRPVB in turbulence gradually converts into a flat-topped distribution with increasing propagation distance and zenith angle, and finally evolves into a Gaussian-like profile. Also, we can find that a PCRPVB with the high topological charges l has the stronger ability of resisting turbulence in comparison with the non-vortex beam. Our work will be useful for free-space optical communications, remote sensing, and the lidar distance measurement.