This paper addresses the challenge of balancing the need for rigorous safety guarantees with the need for responsive and efficient operation in collaborative robot cells to enable flexible, high-throughput automation in human-robot shared workspaces. Specifically, we aim to overcome the limitations of purely reactive or worst-case assumption approaches by considering the stochastic and task-dependent variability of human motion (dynamic obstacles). We highlight that existing learning-based human motion prediction approaches tend to predict worst-case scenarios and struggle to handle prediction uncertainty, resulting in overly conservative planning algorithms. Therefore, we propose a framework called Uncertainty-Aware Predictive Control Barrier Functions (UA-PCBFs), which integrates probabilistic human hand motion prediction with the formal safety guarantees of Control Barrier Functions (CBFs). UA-PCBFs dynamically adjusts safety margins based on human motion uncertainty estimates provided by the prediction module. We validate UA-PCBFs through automated setup experiments with real robot hands and direct human-robot interaction experiments, demonstrating that it outperforms existing HRI architectures in task-related metrics and significantly reduces the number of safe space violations.