This article investigates the unmanned aerial vehicle (UAV)-assisted wireless powered Internet-of-Things system, where a UAV takes off from a data center, flies to each of the ground sensor nodes (SNs) in order to transfer energy and collect data from the SNs, and then returns to the data center. For such a system, an optimization problem is formulated to minimize the average Age of Information (AoI) of the data collected from all ground SNs. Since the average AoI depends on the UAV's trajectory, the time required for energy harvesting (EH) and data collection for each SN, these factors need to be optimized jointly. Moreover, instead of the traditional linear EH model, we employ a nonlinear model because the behavior of the EH circuits is nonlinear by nature. To solve this nonconvex problem, we propose to decompose it into two subproblems, i.e., a joint energy transfer and data collection time allocation problem and a UAV's trajectory planning problem. For the first subproblem, we prove that it is convex and give an optimal solution by using Karush-Kuhn-Tucker (KKT) conditions. This solution is used as the input for the second subproblem, and we solve optimally it by designing dynamic programming (DP) and ant colony (AC) heuristic algorithms. The simulation results show that the DP-based algorithm obtains the minimal average AoI of the system, and the AC-based heuristic finds solutions with near-optimal average AoI. The results also reveal that the average AoI increases as the flying altitude of the UAV increases and linearly with the size of the collected data at each ground SN.