We study a hybrid satellite-terrestrial cognitive network (HSTCN) relying on non-orthogonal multiple access (NOMA) interconnecting a satellite and multiple terrestrial nodes. In this scenario, the long distance communication is achieved by the satellite equipped multiple antennas to send information to a multi-antenna destinations through the base station acting as relay. The secure performance is necessary to study by exploiting the appearance of an eavesdropper attempting to intercept the transmissions from relay to destinations. We explore situation of hardware imperfections in secondary network and deign of multiple antennas need be investigated in term of the physical-layer security by adopting the decode-and-forward (DF) relay strategy. Specifically, we guarantee coverage area by enabling relaying scheme and keep outage probability (OP) performance satisfying required data rates. Moreover, suppose that only the main channels' state information is known while the wiretap channels' state information is unavailable due to the passive eavesdropper, we analyze the secrecy performance in term of intercept probability (IP) of the HSTCN by driving the closed-form expressions of such performance metric. Finally, the presented simulation results show that: 1) The outage behaviors of NOMA-based HSTCN network does not depend on transmit signal to noise ratio (SNR) at source at high SNR; 2) Numerical results show that the such system using higher number of transceiver antennas generally outperform the system with less antennas in terms of OP and IP and reasonable selection of parameters is necessary to remain the secrecy performance of such systems; and 3) By allocating different power levels to tow users, the second user has better secure behavior compared with the first user regardless of other set of satellite links or the number of antennas, which means that the superiority of the second user compared with user the first user in terms of OP and IP are same. INDEX TERMS hybrid satellite-terrestrial cognitive systems, outage probability, Shadowed-Rician fading