Recent dialogue systems rely on turn-based spoken interactions, requiring accurate Automatic Speech Recognition (ASR). Errors in ASR can significantly impact downstream dialogue tasks. To address this, using dialogue context from user and agent interactions for transcribing subsequent utterances has been proposed. This method incorporates the transcription of the user's speech and the agent's response as model input, using the accumulated context generated by each turn. However, this context is susceptible to ASR errors because it is generated by the ASR model in an auto-regressive fashion. Such noisy context can further degrade the benefits of context input, resulting in suboptimal ASR performance. In this paper, we introduce Context Noise Representation Learning (CNRL) to enhance robustness against noisy context, ultimately improving dialogue speech recognition accuracy. To maximize the advantage of context awareness, our approach includes decoder pre-training using text-based dialogue data and noise representation learning for a context encoder. Based on the evaluation of speech dialogues, our method shows superior results compared to baselines. Furthermore, the strength of our approach is highlighted in noisy environments where user speech is barely audible due to real-world noise, relying on contextual information to transcribe the input accurately.