A new specially correlated partially coherent beam named nonuniform multi-Gaussian correlated (NMGC) partially coherent beam is introduced. The correlation functions of such beam in x and y directions are different from each other, i.e., nonuniform correlation function in one direction and multi-Gaussian correlated Schell-model function in the other direction. The propagation properties of an NMGC partially coherent beam in free pace are demonstrated, and we find that the intensity distribution of such beam exhibits self-focusing and self-shifting effect in one direction and self-shaping effect in the other direction on propagation. The correlation-induced self-focusing and self-shaping effect will be useful in some applications, where the high power and shaped laser is required, such as material thermal processing and laser carving.of China under grant no. 11274005 and the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.