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[en] We characterize the non-Gaussian character of two-mode continuous-variable states based on the phase-averaged cumulant approach. We study the dynamics of non-Gaussianity of two-mode hybrid entangled states generated via the cross-Kerr nonlinearity by analyzing the fourth-order phase-averaged cumulant. It is shown that for a case of stronger coherent for one incident light, the non-Gaussianity undergoes some oscillations conditioned on the initial conditions of another incident light. Interestingly, one can simultaneously obtain the two-mode state’s maximum entanglement and highly non-Gaussian correlations by controlling the cross-Kerr interaction. Finally, we discuss the relationship between the entanglement and the variational non-Gaussianity of such states. Graphical abstract: .