Covert throughput maximization for a non-orthogonal multiple access (NOMA)-based visible light covert communication (VLCC) network is investigated. The network consists of a light emitting diode (LED) transmitter, two NOMA users (one public, one covert), and a monitor tasked with detecting any covert transmissions between the LED and the covert user. The transmitter leverages its interaction with the public user to mask the covert communication with the covert user, adopting a random power transmission scheme. This strategy serves to amplify the monitor’s detection uncertainty and significantly enhance the covertness of the VLCC network. Two VLCC scenarios are covered: For the indoor static VLCC scenario where the LED is fixed, subject to the minimum detection error probability of the monitor (covertness constraint) and the outage probability of NOMA users (reliability constraint), the covert throughput is maximized by optimizing the ratio of the LED’s power allocation factor (PAF). For the mobile VLCC scenario where the LED is mounted on an unmanned aerial vehicle (UAV), subject to the constraints of the covertness, reliability and UAV’s flight region, the optimal LED’s PAF ratio and UAV’s location are jointly obtained via a graphical approach. Finally, simulations are carried out to analyze the influence of VLCC parameters on the maximum covert throughput, and results show that compared with benchmark schemes, the proposed scheme can greatly improve the covert throughput.