In this paper, we investigate a non-orthogonal multiple access (NOMA) assisted cognitive satellite-terrestrial network under practical system conditions, such as transceiver hard-ware impairments, channel state information mismatch, imperfect successive interference cancellation and interference noises. Generalized coverage probability formulas for NOMA users in both primary and secondary networks are derived considering the impact of interference temperature constraint. Moreover, the numerical results demonstrate superior outperformance compared to the ones obtained for an orthogonal multiple access scheme. Finally, the derived analytical findings are fully supported by thorough Monte Carlo simulations.