Early-phase dose-finding clinical trials are often subject to the issue of late-onset outcomes. In phase I/II clinical trials, the issue becomes more intractable because toxicity and efficacy can be competing risk outcomes such that the occurrence of the first outcome will terminate the other one. In this paper, we propose a novel Bayesian adaptive phase I/II clinical trial design to address the issue of late-onset competing risk outcomes. We use the continuation-ratio model to characterize the trinomial response outcomes and the cause-specific hazard rate method to model the competing-risk survival outcomes. We treat the late-onset outcomes as missing data and develop a Bayesian data augmentation method to impute the missing data from the observations. We also propose an adaptive dose-finding algorithm to allocate patients and identify the optimal biological dose during the trial. Simulation studies show that the proposed design yields desirable operating characteristics.