The propagation of electromagnetic wave in plasma is one of the long-standing concerns in the field of laser plasma, and it is closely related to the researches of radiation source generation, particle acceleration, and inertial confinement fusion. Recently, the proposal of various schemes for generating intense vortex beams has led to an increasing number of researchers focusing on the interaction between intense vortex beams and plasmas, resulting in significant research progress in various areas, such as particle acceleration, high-order harmonic generation, quasi-static self-generated magnetic fields, and parametric instability. Compared with traditional Gaussian beams, vortex beams, featuring their hollow amplitudes and helical phases, can exhibit novel phenomena during propagating through plasma. In this work, we primarily focus on studying the influence of the propagation process on the wave structure of vortex beams before filamentation occurs. The three-dimensional particle-in-cell simulations show that weakly relativistic vortex beams exhibit wavefront distortion during their propagation in plasma. The distortion degree is closely related to the intensity of the electromagnetic wave and the propagation distance for a given plasma density. This phenomenon is theoretically explained by using a phase correction model that considers the relativistic mass correction of electrons. Additionally, we demonstrate that the wavefront distortion can be compensated for and suppressed by appropriately modulating the initial plasma density, as confirmed by three-dimensional particle simulations. The results of decomposing the wavefront into Laguerre-Gaussian (LG) mode components indicate that the wavefront distortion is primarily caused by high-order p LG modes, and it is independent of other l LG modes. Additionally, we extend the present investigation to the propagation of vortex beams in axially magnetized plasma, where the phase correction model can also effectively explain the occurrence of wavefront distortion. Our work can deepen the understanding of the interaction between plasma and strong vortex beams, and provide some valuable references for designing plasma devices serving as the manipulation of intense vortex beams in future research.