We study the propagation properties of a recently introduced class of structured beams, radially polarized Hermite non-uniformly correlated (RPHNUC) beams, in a turbulent atmosphere using the extended Huygens-Fresnel integral and investigate how the mode order and coherence width play a role in resisting the degradation and depolarization effects of the turbulence. In contrast with conventional vector partially coherent beams (PCBs) with uniform (Schell-model) correlation structure, the interaction of the non-uniform correlation structure and non-uniform polarization gives these beams the ability to self-heal their intensity distribution and polarization over certain propagation ranges in turbulence. These properties suggest that RPIINLIC beams may be useful in a number of applications, in particular optical trapping and free-space optical communications. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement