In this paper, we investigate deep neural network (DNN)-based power allocation policies maximizing the opportunistic rate of a relay-aided cognitive radio network under a quality of service (QoS) constraint protecting the primary transmission. The full-duplex relay performs either Decode-and-Forward (DF) or Compress-and-Forward (CF) and assists the opportunistic communication. The considered primary QoS constraint is expressed in terms of the tolerated primary rate degradation compared to the case of no opportunistic interference. In order to cope with imperfect channel state information (CSI) especially regarding the links to/from the primary network, we propose a self-supervised learning approach that skillfully exploits both perfect and imperfect CSI knowledge within the training phase. Since none of the two relaying schemes is optimal in all system setups (e.g., relative position of the different transmitters, receiver and of the relay), we then propose a novel supervised DNN-based relaying scheme selection. Finally, we extend all these results by proposing a self-supervised DNN-based power allocation policy that is able to generalize over system parameters such as the individual power budget, and the allowed level of primary degradation. Our extensive numerical results on synthetic data demonstrate the effectiveness of our proposed deep learning approaches.