The problem of filling in missing or damaged wavelet coefficients is considered in this paper. Chan, Shen, and Zhou have proposed two total variation (TV) wavelet inpainting models to solve this problem. The main benefit of TV model is that it can keep the edges very well, but this method suffers from the staircase effect. To overcome this defect, we analyze the physical characteristics of TV model and p-Laplace operator in local coordinates, and explain that diffusion performance of p-Laplace is superior to that of TV model in essence. Afterwards, an inpainting model based on p-Laplace operator for damaged wavelet coefficients is presented. This new model can effectively reduce the staircase effect in TV model whereas it can still keep sharp edges as well as TV model. Experiment results show that better inpaingting quality can be achieved with much less computing time with our model.