Audio-visual speech recognition (AVSR) incorporates auditory and visual modalities to improve recognition accuracy, particularly in noisy environments where audio-only speech systems are insufficient. While previous research has largely addressed audio disruptions, few studies have dealt with visual corruptions, e.g., lip occlusions or blurred videos, which are also detrimental. To address this real-world challenge, we propose CAV2vec, a novel self-supervised speech representation learning framework particularly designed to handle audio-visual joint corruption. CAV2vec employs a self-distillation approach with a corrupted prediction task, where the student model learns to predict clean targets, generated by the teacher model, with corrupted input frames. Specifically, we suggest a unimodal multi-task learning, which distills cross-modal knowledge and aligns the corrupted modalities, by predicting clean audio targets with corrupted videos, and clean video targets with corrupted audios. This strategy mitigates the dispersion in the representation space caused by corrupted modalities, leading to more reliable and robust audio-visual fusion. Our experiments on robust AVSR benchmarks demonstrate that the corrupted representation learning method significantly enhances recognition accuracy across generalized environments involving various types of corruption.