Human brain possesses the ability to effectively focus on important environmental components, which enhances perception, learning, reasoning, and decision-making. Inspired by this cognitive mechanism, we introduced a novel concept termed relevance for Human-Robot Collaboration (HRC). Relevance is a dimensionality reduction process that incorporates a continuously operating perception module, evaluates cue sufficiency within the scene, and applies a flexible formulation and computation framework. In this paper, we present an enhanced two-loop framework that integrates real-time and asynchronous processing to quantify relevance and leverage it for safer and more efficient human-robot collaboration (HRC). The two-loop framework integrates an asynchronous loop, which leverages LLM world knowledge to quantify relevance, and a real-time loop, which performs scene understanding, human intent prediction, and decision-making based on relevance. HRC decision-making is enhanced by a relevance-based task allocation method, as well as a motion generation and collision avoidance approach that incorporates human trajectory prediction. Simulations and experiments show that our methodology for relevance quantification can accurately and robustly predict the human objective and relevance, with an average accuracy of up to 0.90 for objective prediction and up to 0.96 for relevance prediction. Moreover, our motion generation methodology reduces collision cases by 63.76% and collision frames by 44.74% when compared with a state-of-the-art (SOTA) collision avoidance method. Our framework and methodologies, with relevance, guide the robot on how to best assist humans and generate safer and more efficient actions for HRC.