In this paper, we propose a novel Hybrid-Hierarchical spatial-aerial-Terrestrial Edge-Centric (H2TEC) for the space-air integrated Internet of Things (IoT) networks. (H2TEC) comprises unmanned aerial vehicles (UAVs) that act as mobile fog nodes to provide the required services for terminal nodes (TNs) in cooperation with the satellites. TNs in (H2TEC) offload their generated tasks to the UAVs for further processing. Due to the limited energy budget of TNs, a novel task allocation protocol, named TOP, is proposed to minimize the energy consumption of TNs while guaranteeing the outage probability and network reliability for which the transmission rate of TNs is optimized. TOP also takes advantage of the energy harvesting by which the low earth orbit satellites transfer energy to the UAVs when the remaining energy of the UAVs is below a predefined threshold. To this end, the harvested power of the UAVs is optimized alongside the corresponding harvesting time so that the UAVs can improve the network throughput via processing more bits. Numerical results reveal that TOP outperforms the baseline method in critical situations that more power is required to process the task. It is also found that even in such situations, the energy harvesting mechanism provided in the TOP yields a more efficient network throughput.