Unmanned aerial vehicles (UAVs) can help facilitate cost-effective and flexible service provisioning in future smart cities. Nevertheless, UAV applications generally suffer severe flight time limitations due to constrained onboard battery capacity, causing a necessity of frequent battery recharging or replacement when performing persistent missions. Utilizing wireless mobile chargers, such as vehicles with wireless charging equipment for on-demand self-recharging has been envisioned as a promising solution to address this issue. In this article, we present a comprehensive study of \underline{v}ehicle-assisted \underline{w}ireless rechargeable \underline{U}AV \underline{n}etworks (VWUNs) to promote on-demand, secure, and efficient UAV recharging services. Specifically, we first discuss the opportunities and challenges of deploying VWUNs and review state-of-the-art solutions in this field. We then propose a secure and privacy-preserving VWUN framework for UAVs and ground vehicles based on differential privacy (DP). Within this framework, an online double auction mechanism is developed for optimal charging scheduling, and a two-phase DP algorithm is devised to preserve the sensitive bidding and energy trading information of participants. Experimental results demonstrate that the proposed framework can effectively enhance charging efficiency and security. Finally, we outline promising directions for future research in this emerging field.