Ambient Backscatter Communication (AmBC) is an innovative approach to wireless communication, leveraging ambient radio frequency (RF) signals instead of self-generated ones. This unique feature positions AmBC as an appealing choice for future Internet-of-Things (IoT) applications, especially when considering the challenges associated with energy and spectrum resources. Additionally, cognitive radio (CR) is considered a leading technology to meet the spectrum demands of wireless networks by engaging in spectrum sharing with licensed users, also known as primary users (PUs). In this work, we investigate an underlay network of backscatter devices (BDs) based on CR that uses non-orthogonal multiple access (NOMA) for communication with the secondary network backscatter receiver (SBR). Furthermore, by fulfilling the interference and quality of service (QoS) constraints of the primary user and BDs, respectively, and optimizing the power reflection coefficients of BDs, an optimization problem is formulated and solved to maximize the sum rate of the proposed network. According to the simulation results, the suggested NOMA-based network outperforms the baseline conventional orthogonal multiple access (OMA) based networks in terms of sum rate performance by a margin of 21.86% to 27.74%, depending upon transmit power levels.