Fifth-generation (5G) networks offer high-speed data transmission with low latency, increased base station volume, improved quality of service (QoS), and massive multiple-input-multiple-output (M-MIMO) channels compared to 4G long-term evolution (LTE) networks. However, the COVID-19 pandemic has disrupted the achievement of mobility and handover (HO) in 5G networks due to significant changes in intelligent devices and high-definition (HD) multimedia applications. Consequently, the current cellular network faces challenges in propagating high-capacity data with improved speed, QoS, latency, and efficient HO and mobility management. This comprehensive survey paper specifically focuses on HO and mobility management issues within 5G heterogeneous networks (HetNets). The paper thoroughly examines the existing literature and investigates key performance indicators (KPIs) and solutions for HO and mobility-related challenges while considering applied standards. Additionally, it evaluates the performance of current models in addressing HO and mobility management issues, taking into account factors such as energy efficiency, reliability, latency, and scalability. Finally, this paper identifies significant challenges associated with HO and mobility management in existing research models and provides detailed evaluations of their solutions along with recommendations for future research.