Unmanned aerial vehicle (UAV)-assisted communication is a promising technology in future wireless communication networks. UAVs can not only help offload data traffic from ground base stations (GBSs) but also improve the Quality of Service (QoS) of cell-edge users (CEUs). In this article, we consider the enhancement of cell-edge communications through a mobile relay, i.e., UAV, in multicell networks. During each transmission period, GBSs first send data to the UAV, and then the UAV forwards its received data to CEUs according to a certain association strategy. In order to maximize the sum rate of all CEUs, we jointly optimize the UAV mobility management, including trajectory, velocity, and acceleration, and association strategy of CEUs to the UAV, subject to minimum rate requirements of CEUs, mobility constraints of the UAV, and causal buffer constraints in practice. To address the mixed-integer nonconvex problem, we transform it into two convex subproblems by applying tight bounds and relaxations. An iterative algorithm is proposed to solve the two subproblems in an alternating manner. Numerical results show that the proposed algorithm achieves higher rates of CEUs as compared with the existing benchmark schemes.