Increasing complexity and availability demands of traditional software systems have derived the need for self-adaptive systems. Self-adaptive systems automatically adjust their behavior at runtime to meet objectives. However, these systems accompany additional challenges specially in terms of security. These challenges include interaction with non-trusted entities, widening of attack surface due to multichannel communication, risk of unauthorized and malicious access to operating environment, and potential exploitation of adaptive mechanisms by attackers. Despite the growing adoption of these systems in avionics, comprehensive study on their cybersecurity assessment is missing. To fill the gap, this study discusses various security vulnerabilities introduced with self-adaptivity in avionics. The aim of the study is to use the Plug&Fly Avionics Platform (PAFA) as a case study, and analyze the impact of self-adaptivity in exposing the system to a wide range of cyber threats such as, faulty decision making, compromised reconfigurations, and so on. Besides detailed analysis of security vulnerabilities introduced with self-adaptivity, the paper also proposes some novel mitigation strategies and demonstrate their effectiveness in achieving a cybersecure adaptive system in the context of PAFA. These findings aim to provide solid basis in future research and development towards securing self-adaptive systems, ensuring both their reliability and safety in complex operational environments.