In unmanned aerial vehicle(UAV) communication networks, trajectory planning and resource allocation(TPRA) under channel prediction obtains great attention due to its significant energy-saving of UAVs and user quality of service(QoS) gains. These potentials are primarily demonstrated under the assumption of perfect channel prediction. However, due to the rapid-varying features of air-to-ground channels,it is difficult to avoid the random channel prediction error(CPE), which may deteriorate the performance of TPRA. In this paper, we investigate the problem of energy-efficient TPRA considering random CPE. The problem is formulated as a mixed-integer non-convex optimization with chance constraints, which ensures that QoS is robust to random CPE. To solve it, we first transform the chance constraints into deterministic forms, which are further proved to be convex constraints by using the characteristic of quasiconvex functions.Then, we design a modified successive convex approximation algorithm to iteratively achieve the optimal solution. To cater to the high-speed movement of UAVs, a low-complexity heuristic online algorithm is tailored. Specifically, we first relax the QoS constraints to find a feasible initial point and iteratively tighten the lower bound of QoS constraints to obtain a suboptimal solution. Simulation results show that the proposed algorithm can improve energy efficiency compared with the algorithm without prediction.