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Cumulative damage processes for discrete degrading systems caused by stochastic shocks have received increasing attention. However, the degradation data is often subject to incomplete observation or censoring due to imperfect monitoring instruments or inspection strategies. In such situations, the arrival time of stochastic shocks cannot be accurately recorded. This makes a challenging task for parameter estimation and remaining useful lifetime (RUL) prognosis based on the cumulative damage process. Motivated by this fact, this article proposed a new Bayesian data augmentation approach to overcome the obstacles caused by incomplete observation. First, the proposal distribution for the occurrence number of shocks in the cumulative damage process subject to interval censoring is derived. Second, an effective data augmentation for the missing shock time is proposed by integrating rejection sampling and the order statistics technique. On this basis, multiple imputation is extended to discrete non-Gaussian stochastic processes subject to interval censoring. Finally, an RUL prognosis method of a system subject to the cumulative damage process is derived based on the Monte Carlo method. And a real case study on the head wear problem is applied to illustrate the superiority of the proposed approach, and the results show that the proposed approach can improve the estimation accuracy of RUL compared with the existing method. [ABSTRACT FROM AUTHOR]