Incorporating automatically predicted human feedback into the process of training generative models has attracted substantial recent interest, while feedback at inference time has received less attention. The typical feedback at training time, i.e., preferences of choice given two samples, does not naturally transfer to the inference phase. We introduce a novel type of feedback -- caption reformulations -- and train models to mimic reformulation feedback based on human annotations. Our method does not require training the image captioning model itself, thereby demanding substantially less computational effort. We experiment with two types of reformulation feedback: first, we collect a dataset of human reformulations that correct errors in the generated captions. We find that incorporating reformulation models trained on this data into the inference phase of existing image captioning models results in improved captions, especially when the original captions are of low quality. We apply our method to non-English image captioning, a domain where robust models are less prevalent, and gain substantial improvement. Second, we apply reformulations to style transfer. Quantitative evaluations reveal state-of-the-art performance on German image captioning and English style transfer, while human validation with a detailed comparative framework exposes the specific axes of improvement.