In fully sampled cardiac MR (CMR) acquisitions, motion can lead to corruption of k-space lines, which can result in artefacts in the reconstructed images. In this paper, we propose a method to automatically detect and correct motion-related artefacts in CMR acquisitions during reconstruction from k-space data. Our correction method is inspired by work on undersampled CMR reconstruction, and uses deep learning to optimize a data-consistency term for under-sampled k-space reconstruction. Our main methodological contribution is the addition of a detection network to classify motion-corrupted k-space lines to convert the problem of artefact correction to a problem of reconstruction using the data consistency term. We train our network to automatically correct for motion-related artefacts using synthetically corrupted cine CMR k-space data as well as uncorrupted CMR images. Using a test set of 50 2D+time cine CMR datasets from the UK Biobank, we achieve good image quality in the presence of synthetic motion artefacts. We quantitatively compare our method with a variety of techniques for recovering good image quality and showcase better performance compared to state of the art denoising techniques with a PSNR of 37.1. Moreover, we show that our method preserves the quality of uncorrupted images and therefore can be also utilized as a general image reconstruction algorithm.