Semi-supervised learning in automatic speech recognition (ASR) typically relies on pseudo-labeling, which often suffers from confirmation bias and error accumulation due to noisy supervision. To address this limitation, we propose ReHear, a framework for iterative pseudo-label refinement that integrates an instruction-tuned, audio-aware large language model (LLM) into the self-training loop. Unlike conventional text-based correctors, our approach conditions the LLM on both the ASR hypothesis and the source audio, allowing it to recover phonetically accurate transcripts even from severe recognition errors. These refined pseudo-labels serve as high-fidelity targets for fine-tuning the ASR model in an iterative cycle. Experimental results across diverse benchmarks demonstrate that ReHear effectively mitigates error propagation, consistently outperforming both supervised and pseudo-labeling baselines.