Training automatic speech recognition (ASR) systems requires large amounts of well-curated paired data. However, human annotators usually perform "non-verbatim" transcription, which can result in poorly trained models. In this paper, we propose Omni-temporal Classification (OTC), a novel training criterion that explicitly incorporates label uncertainties originating from such weak supervision. This allows the model to effectively learn speech-text alignments while accommodating errors present in the training transcripts. OTC extends the conventional CTC objective for imperfect transcripts by leveraging weighted finite state transducers. Through experiments conducted on the LibriSpeech and LibriVox datasets, we demonstrate that training ASR models with OTC avoids performance degradation even with transcripts containing up to 70% errors, a scenario where CTC models fail completely. Our implementation is available at https://github.com/k2-fsa/icefall.