This study proposes an integrated system that combines energy harvesting (EH) enabled unmanned aerial vehicles (UAVs) with non-orthogonal multiple access (NOMA) to enhance communication system performance within a cellular network. Addressing the limitations of existing analyses that often assume an infinite blocklength scenario, we explore EH-enabled UAV-NOMA systems within a cellular framework under a finite blocklength (FBL) scenario. The study investigates the complex interactions and advantages resulting from the integration of EH, NOMA, and UAV technologies, aiming to assess whether EH can sustain communication within this framework. The network model considers base stations (BSs), UAVs, and terrestrial devices distributed with independent Poisson point processes (PPPs) over a large area. In this network, BSs employ NOMA to serve cell center devices directly, while cell edge devices, which are nor in direct contact with BS, are served via simultaneous wireless information and power transfer (SWIPT) enabled UAVs. The study derives metrics including joint harvesting and decoding probability for a randomly selected UAV, coverage probability (CP) for cell devices, and end-to-end block error rate (BLER) probabilities for typical device pairs. The findings demonstrate that the proposed scheme effectively supplies all the necessary transmit power for communication purposes through EH, achieving reasonable reliability. Additionally, the study highlights the importance of considering a combination of blocklengths from different phases to achieve optimal performance, rather than solely relying on an increment in blocklength. Finally, the effects of parameter variations on network performance are examined.