Bayesian network structure learning is a useful tool for elucidation of regulatory structures of biomolecular pathways. The approach however is limited by its acyclicity constraint, a problematic one in the cycle-containing biological domain. Here, we introduce a novel method for modeling cyclic pathways in biology, by employing our newly introduced Generalized Bayesian Networks (GBNs). Our novel algorithm enables cyclic structure learning while employing biologically relevant data, as it extends our cycle-learning algorithm to permit learning with singly perturbed samples. We present theoretical arguments as well as structure learning results from realistic, simulated data of a biological system. We also present results from a real world dataset, involving signaling pathways in T-cells.