When Unmanned Aerial Vehicles (UAVs) perform high-precision communication tasks, such as searching for users and providing emergency coverage, positioning errors between base stations and users make it challenging to deploy trajectory planning algorithms. To address these challenges caused by position errors, a framework was proposed to compensate it by Channel Knowledge Map (CKM), which stores channel state information (CSI). By taking the positions with errors as input, the generated CKM could give a prediction of signal attenuation which is close to true positions. Based on that, the predictions are utilized to calculate the received power and a PPO-based algorithm is applied to optimize the compensation. After training, the framework is able to find a strategy that minimize the flight time under communication constraints and positioning error. Besides, the confidence interval is calculated to assist the allocation of power and the update of CKM is studied to adapt to the dynamic environment. Simulation results show the robustness of CKM to positioning error and environmental changes, and the superiority of CKM-assisted UAV communication design.