Stochastic reaction network modelling is widely utilized to describe the probabilistic dynamics of biochemical systems in general, and gene interaction networks in particular. The statistical analysis of the response of these systems to perturbation inputs is typically dependent on specific perturbation models. Motivated by reporter gene systems, widely utilized in biology to monitor gene activity in individual cells, we address the analysis of reaction networks with state-affine rates in presence of an input process. We develop a generalization of the so-called moment equations that precisely accounts for the first- and second-order moments of arbitrary inputs without the need for a model of the input process, as well as spectral relationships between the network input and state. We then apply these results to develop a method for the reconstruction of the autocovariance function of gene activity from reporter gene population-snapshot data, a crucial step toward the investigation of gene regulation, and demonstrate its performance on a simulated case study.