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In remote and inaccessible areas, the traffic request for Internet-of-Things (IoT) terminals is growing. This paper proposes a cognitive satellite-aerial network (CSAN) to provide sufficient access services. The proposed CSAN consists of the primary beam-hopping (BH) satellite and secondary aerial-based station (ABS) systems. Since the two systems share spectrums, co-channel interference (CCI) between the two systems is complicated, and the quality of service (QoS) is seriously degraded. To improve the QoS, the dynamic BH (DBH) pattern, ABS access in the time domain, and ABS power control in the power domain are studied. First, based on the sparsity of the DBH pattern, the greedy quick tracking (GAT) algorithm is proposed to design the DBH pattern quickly. Then, subject to the DBH pattern, a greedy access monitor (GAM) algorithm is determined for timely ABS access and power control. Since each ABS only serves terminals within a suitable distance, the placement and terminal cluster of multi-ABSs in the space domain are required to ensure full terminal coverage. Thus, the mutual selection K algorithm is proposed to save required ABS numbers and improve service fairness among terminal clusters. Simulation results demonstrate the efficacy of time-space-power allocation for enhanced IoT-terminal services in the proposed CSAN.