Generative error correction (GER) with large language models (LLMs) has emerged as an effective post-processing approach to improve automatic speech recognition (ASR) performance. However, it often struggles with rare or domain-specific words due to limited training data. Furthermore, existing LLM-based GER approaches primarily rely on textual information, neglecting phonetic cues, which leads to over-correction. To address these issues, we propose a novel LLM-based GER approach that targets rare words and incorporates phonetic information. First, we generate synthetic data to contain rare words for fine-tuning the GER model. Second, we integrate ASR's N-best hypotheses along with phonetic context to mitigate over-correction. Experimental results show that our method not only improves the correction of rare words but also reduces the WER and CER across both English and Japanese datasets.