The increasing importance of terminal privacy in the Unmanned Aerial Vehicle (UAV) network has led to a growing recognition of the crucial role of authentication technology in UAV network security. However, traditional authentication approaches are vulnerable due to the transmission of identity information between UAVs and cryptographic paradigm management centers over a public channel. These vulnerabilities include brute-force attacks, single point of failure, and information leakage. Blockchain, as a decentralized distributed ledger with blockchain storage, tamper-proof, secure, and trustworthy features, can solve problems such as single-point-of-failure and trust issues, while the hidden communication in the physical layer can effectively resist information leakage and violent attacks. In this paper, we propose a lightweight UAV network authentication mechanism that leverages blockchain and covert communication, where the identity information is transmitted as covert tags carried by normal modulated signals. In addition, a weight-based Practical Byzantine Fault-Tolerant (wPBFT) consensus protocol is devised, where the weights are determined by the channel states of UAVs and the outcomes of past authentication scenarios. Simulation results demonstrate that the proposed mechanism outperforms traditional benchmarks in terms of security and robustness, particularly under conditions of low Signal-to-Noise Ratio (SNR) and short tag length.