In this paper, we introduce a general model for jointly modelling the nodal heterogeneity and covariates in weighted or unweighted bipartite networks, which contains two different types of nodes. The model has a degree heterogeneity parameter for each node and a fixed-dimensional regression coefficient for the covariates. We use the method of moments to estimate the unknown parameters. When the model belongs to the exponential family of distributions, the moment estimator is identical to the maximum likelihood estimator. We show the uniform consistency of the moment estimator, when the number of actors and the number of events both go to infinity under some conditions. Further, we derive an asymptotic representation of the moment estimator, which leads to their asymptotic normal distributions under some conditions. We present two applications to illustrate the unified results. Numerical simulations and a real-data analysis demonstrates our theoretical findings.