The conventional partially coherent beam has a Gaussian correlated Schell-model function. In 2007, Gori and Santarsiero[Gori F, Santarsiero M 2007 Opt. Lett. 32 3531] discussed the sufficient condition for devising a genuine correlation function of a partially coherent beam. Since then, a variety of partially coherent beams with nonconventional correlation functions, such as nonuniform correlated beam, Hermite-Gaussian correlated beam, Laguerre-Gaussian correlated beam and beam with optical coherence lattices, have been introduced, and such beams display many extraordinary propagation properties, such as self-focusing, self-shifting, self-splitting, self-shaping and periodicity reciprocity, and they have useful applications in many areas, such as free-space optical communication, particle trapping, image transmission and optical encryption.In most of previous studies, the correlation function of the partially coherent beam was assumed to be isotropic. In this paper, we introduce a new kind of partially coherent beam with anisotropic correlation function, which is named nonuniform Laguerre-Gaussian correlated（NLGC) beam. The NLGC beam has a nonuniform correlated function in the x-direction and Laguerre-Gaussian correlated Schell-model function in the y-direction. Furthermore, we explore the propagation properties of the NLGC beam in free space and in turbulent atmosphere comparatively with the help of the extended Huygens-Fresnel integral. In free space, it is found that the intensity distribution of the NLGC beam displays self-focusing and self-shifting behaviors in the x-direction and self-splitting properties in the y-direction during its propagation, which may be useful for particle trapping, and the distribution of the degree of coherence also varies during its propagation. In turbulent atmosphere, the NLGC beam displays similar propagation properties at short propagation distance because the influence of turbulence can be neglected, while with the further increase of the propagation distance, the influence of turbulence accumulates and both the intensity distribution and the degree of coherence distribution evolve into Gaussian profiles. We also find that the evolution properties of the intensity distribution and the degree of coherence are closely related to the mode order m of the correlation function, e.g. the intensity distribution and the degree of coherence distribution evolve into Gaussian profiles more slowly as the mode order m increases, which means that the NLGC beam with larger m is less affected by turbulence, which may be useful in free-space optical communication.Our results clearly show that modulating the correlation function of a partially coherent beam provides a novel way of manipulating its propagation properties, and will be useful in many applications, where light beam is required to possess a prescribed beam profile and controlled propagation properties. In this paper, only the NLGC beam is treated theoretically, and such a beam deserves further experimental investigation.