In this paper, we study covert communications in an unmanned aerial vehicle (UAV)-enabled network, where a UAV transmits information to multiple ground users (GUs) without being detected by a hidden detector. Considering fairness issue, we aim at maximizing the minimum throughput among GUs by jointly optimizing the UAV’s trajectory, transmit power and power allocation coefficient, under UAV mobility and covertness constraints. On the one hand, according to the covertness constraint, the maximal transmit power is characterized as a close form expression of UAV’s position. On the other hand, the optimal UAV trajectory structure is characterized as a successive-hover-and-fly (SHF) structure. Following the two fundamental characterizations, we first transform the original problem to a joint trajectory and power allocation design one and then it is reformulated to another one addressing only a limited number of hovering points, corresponding hovering durations, turning points and allocation coefficient. Although being still non-convex, the new problem is efficiently solved via applying the sequential convex programming (SCP) method. Namely, by introducing a series of tight concave function in each iteration, we can solve a series of convex problems iteratively to make the trajectory converge to a high-quality solution. Numerical results confirm the convergence of our approach and show the high performance comparing with benchmark.