This work considers unmanned aerial vehicle (UAV) networks for collecting data covertly from ground users. The full-duplex UAV intends to gather critical information from a scheduled user (SU) through wireless communication and generate artificial noise (AN) with random transmit power in order to ensure a negligible probability of the SUs transmission being detected by the unscheduled users (USUs). To enhance the system performance, we jointly design the UAVs trajectory and its maximum AN transmit power together with the user scheduling strategy subject to practical constraints, e.g., a covertness constraint, which is explicitly determined by analyzing each USUs detection performance, and a binary constraint induced by user scheduling. The formulated design problem is a mixed-integer non-convex optimization problem, which is challenging to solve directly, but tackled by our developed penalty successive convex approximation (P-SCA) scheme. An efficient UAV trajectory initialization is also presented based on the Successive Hover-and-Fly (SHAF) trajectory, which also serves as a benchmark scheme. Our examination shows the developed P-SCA scheme significantly outperforms the benchmark scheme in terms of achieving a higher max-min average transmission rate from all the SUs to the UAV.