We investigate whether Graph Convolutional Neural Networks (GCNNs) may benefit from incorporating information conveyed by a state-of-the-art graph kernel in the learning process. We propose a GCNN architecture and a training procedure based on multi-task learning, where we provide supervision not only from the graph labels, but also from the kernel to each layer of the network, achieving state-of-the-art performances on many real-world datasets. We conduct an ablation study to analyze the impact on the predictive performances of each part of our proposal, including a simplified version of our multi-task learning formulation that can, in principle, be applied with a broad family of graph embeddings. Finally, we study how to improve the performance of a model considering graphs coming from related datasets into the training procedure in a semi-supervised learning fashion.