Specular reflections (SRs) are highlight artifacts commonly found in endoscopy videos that can severely disrupt a surgeon's observation and judgment. Despite numerous attempts to restore SR, existing methods are inefficient and time consuming and can lead to false clinical interpretations. Therefore, we propose the first complete deep-learning solution, SpecReFlow, to detect and restore SR regions from endoscopy video with spatial and temporal coherence. SpecReFlow consists of three stages: (1)an image preprocessing stage to enhance contrast, (2)a detection stage to indicate where the SR region is present, and (3)a restoration stage in which we replace SR pixels with an accurate underlying tissue structure. Our restoration approach uses optical flow to seamlessly propagate color and structure from other frames of the endoscopy video. Comprehensive quantitative and qualitative tests for each stage reveal that our SpecReFlow solution performs better than previous detection and restoration methods. Our detection stage achieves a Dice score of 82.8% and a sensitivity of 94.6%, and our restoration stage successfully incorporates temporal information with spatial information for more accurate restorations than existing techniques. SpecReFlow is a first-of-its-kind solution that combines temporal and spatial information for effective detection and restoration of SR regions, surpassing previous methods relying on single-frame spatial information. Future work will look to optimizing SpecReFlow for real-time applications. SpecReFlow is a software-only solution for restoring image content lost due to SR, making it readily deployable in existing clinical settings to improve endoscopy video quality for accurate diagnosis and treatment.