In this work, we construct a fully connected multi-layer photonic spiking neural network (PSNN) based on excitable vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSELs-SA) equipped with the modified supervised backpropagation (BP) algorithm. Different from the traditional BP algorithm, our proposed modified BP algorithm is based on optical spikes, and its activation function exploits the excitable characteristic of VCSELs-SA. It is demonstrated that this BP PSNN can be competent in linearly inseparable problems by correctly solving the exclusive or (XOR) problem. Thus, a three-layer fully connected BP PSNN is designed to perform the prediction of Human immunodeficiency virus-1 protease (HIV-1 PR) cleavage sites and achieves 91.95% classification accuracy with the modified BP algorithm, which could be a useful tool in helping to find effective inhibitors of HIV PR to fight against acquired immunodeficiency syndrome (AIDS). These results demonstrate the validity of our modified BP algorithm in dealing with nonlinear classification problems in PSNNs, which is helpful for supervised learning in the photonic neuromorphic computing architecture for future neuromorphic applications.