<div>In spatio-temporal data analysis, the problem of non-separable space-time covariance functions is important and hard to deal with. Most of the famous constructions of these covariance functions are fully symmetric, which is inappropriate in many spatiotemporal processes. The Non-Fully Symmetric Space-Time (NFSST) Matern model by Zhang, T. and Zhang, H. (2015) provides a way to construct a non-fully symmetric non-separable space-time correlation function from marginal spatial and temporal Matern correlation functions.</div><div>In this work we use the relationship between the spatial Matern and temporal Cauchy correlation functions and their spectral densities, and provide a modification to their Bochner’s representation by including a space-time interaction term. Thus we can construct a non-fully symmetric space-time Matern-Cauchy model, from any given marginal spatial Matern and marginal temporal Cauchy correlation functions. We are able to perform computation and parameter estimate on this family, using the Taylor expansion of the correlation functions. This model has attractive properties: it has much faster estimation compared with NFSST Matern model when the spatio-temporal data is large; it enables the existence of temporal long-range dependence (LRD), adding substantially to the flexibility of marginal correlation function in the time domain. Several spatio-temporal meteorological data sets are studied using our model, including one case with temporal LRD.</div>