Head motion occurring during brain positron emission tomography images acquisition leads to a decrease in image quality and induces quantification errors. We have previously introduced a Deep Learning Head Motion Correction (DL-HMC) method based on supervised learning of gold-standard Polaris Vicra motion tracking device and showed the potential of this method. In this study, we upgrade our network to a multi-task architecture in order to include image appearance prediction in the learning process. This multi-task Deep Learning Head Motion Correction (mtDL-HMC) model was trained on 21 subjects and showed enhanced motion prediction performance compared to our previous DL-HMC method on both quantitative and qualitative results for 5 testing subjects. We also evaluate the trustworthiness of network predictions by performing Monte Carlo Dropout at inference on testing subjects. We discard the data associated with a great motion prediction uncertainty and show that this does not harm the quality of reconstructed images, and can even improve it.