To develop an efficient motion-compensated reconstruction technique for free-breathing cardiac magnetic resonance imaging (MRI) that allows high-quality images to be reconstructed from multiple undersampled single-shot acquisitions. The proposed method is a joint image reconstruction and motion correction method consisting of several steps, including a non-rigid motion extraction and a motion-compensated reconstruction. The reconstruction includes a denoising with the Beltrami regularization, which offers an ideal compromise between feature preservation and staircasing reduction. Results were assessed in simulation, phantom and volunteer experiments. The proposed joint image reconstruction and motion correction method exhibits visible quality improvement over previous methods while reconstructing sharper edges. Moreover, when the acceleration factor increases, standard methods show blurry results while the proposed method preserves image quality. The method was applied to free-breathing single-shot cardiac MRI, successfully achieving high image quality and higher spatial resolution than conventional segmented methods, with the potential to offer high-quality delayed enhancement scans in challenging patients.