A new type of power-exponent-phase vortex-like beams with both quadratic and cubic azimuthal phase gradients is investigated in this work. The intensity and orbital angular momentum (OAM) density distributions are noticeably different when the phase gradient increases or decreases along the azimuth angle, while the orthogonality and total OAM remain constant. The characteristics of the optical field undergo a significant change when the phase shifts from linear to nonlinear, with the variation of the power index having little impact on the beam characteristics under nonlinear phase conditions. These characteristics provide new ideas for applications such as particle manipulation, optical communications, and OAM encryption.