Yes / Non-terrestrial Network (NTN), utilizing highaltitude platforms (HAP)-based free-space optical (FSO) backhaul and unmanned aerial vehicles (UAV) for last-mile access, is a feasible and promising architecture to achieve high data rate and seamless network coverage in the future 6G era. Effective resource allocation emerges as a pivotal concern for such networks. This paper addresses the data allocation issue for FSO backhaul from the HAP to multiple UAV-mounted base stations (BSs) under the constraints of ground users’ requested data rates. We introduce frame allocation schemes (FAS), including rate adaptation with constraints (RAC)- and rate/power adaptation (RPA)-aided FAS. The key idea of these schemes is to allocate data frames effectively based on UAV’s turbulence channel conditions, which aims to (i) guarantee the quality of services (QoS), (ii) retain both latency and throughput fairness, and (iii) minimize the transmitted power. Furthermore, the performance of these schemes is also analyzed under the impact of imperfect channel state information (CSI). We newly derive the channel probability density function (PDF) and the cumulative density function (CDF), considering the imperfect CSI due to channel estimation and quantization errors. Capitalizing on the derived PDF and CDF, different performance metrics are analytically obtained, incorporating combined effects of cloud coverage, transceiver misalignment, Fisher-Snedecor F turbulence, and angle-of-arrival (AoA) fluctuations. Numerical results demonstrate the effectiveness of our design proposals over the state-of-the-art. Finally, Monte Carlo simulations are employed to validate the analysis.