Image reconstuction for Positron Emission Tomography (PET) can be modeled as an inverse problem that involves a convolution operator in addition to the Radon transform. Deep learning based methods recently showed convincing results for tomographic reconstruction or for blind deconvolution; however to the best of our knowledge, there is no approach that combines both of these tasks into a unique framework. We propose a neural network that solves this joint task thanks to a hybrid method that includes both supervised and self-supervised learning. We show on simulations that our network significantly outperforms a direct approach using a UNET.