In speech recognition applications, it is important to recognize context-specific rare words, such as proper nouns. Tree-constrained Pointer Generator (TCPGen) has shown promise for this purpose, which efficiently biases such words with a prefix tree. While the original TCPGen relies on grapheme-based encoding, we propose extending it with phoneme-aware encoding to better recognize words of unusual pronunciations. As TCPGen handles biasing words as subword units, we propose obtaining subword-level phoneme-aware encoding by using alignment between phonemes and subwords. Furthermore, we propose injecting phoneme-level predictions from CTC into queries of TCPGen so that the model better interprets the phoneme-aware encodings. We conducted ASR experiments with TCPGen for RNN transducer. We observed that proposed phoneme-aware encoding outperformed ordinary grapheme-based encoding on both the English LibriSpeech and Japanese CSJ datasets, demonstrating the robustness of our approach across linguistically diverse languages.