LLMs have significantly advanced Human-Robot Interaction (HRI) by enabling robots to engage in open-ended, context-aware dialogue. However, LLMs in HRI often give exhaustive responses, leading to information overload, increased cognitive burden, and prolonged task completion, particularly in dynamic scenarios. To address these limitations, we propose the Context-Aware Adaptive LLM Response Strategy for Human-Robot Interaction (CARS-HRI). CARS-HRI integrates multimodal inputs-user gaze, speech features, scene understanding, and task status-into a novel User Cognitive State Assessment Module. This module dynamically predicts the user's cognitive load, task uncertainty, and attentional focus. Based on this assessment, CARS-HRI employs dynamic prompt engineering for an LLM, modulating response length, information density, and guidance style. Experiments with an ambiguous task showed CARS-HRI reduced user cognitive load and significantly shortened task completion compared to a baseline LLM. It also optimized robot response length while maintaining or enhancing user confidence, task execution, and system usability. These findings underscore CARS-HRI's potential to foster more natural, efficient, and cognitively ergonomic bidirectional human-robot interactions.