In this paper, we study radio frequency (RF)-powered underlay cognitive radio networks (CRNs) with power-domain non-orthogonal multiple access (NOMA). In these networks, by using the harvest-then-transmit (HTT) mode, secondary transmitters (STs) can use the harvested energy to simultaneously transmit data based on power-domain NOMA. However, in this mode, the throughput of the secondary system heavily depends not only on the harvested energy, but also on the stringent interference threshold imposed by the primary users. Furthermore, ambient backscatter communication (ABC) has been introduced as a promising technique which enables STs to transmit information by modulating and reflecting ambient RF signals. Therefore, it has the potentiality to be integrated into the RF-powered underlay CR-NOMA networks to improve the throughput of the secondary system. In these networks, each ST works on either the HTT mode or the ABC mode, but not simultaneously. In order to meet the interference constraint of the primary users, STs control their transmit power by finding the appropriate tradeoff between the HTT mode and the ABC mode. We formulate an optimization problem with the goal of achieving the maximum throughput by finding the optimal time resource allocation between the HTT mode and the ABC mode under the strict transmit power constraint at STs. Then the Lagrangian multiplier iterative algorithm is adopted to solve this optimization problem. Simulation results demonstrate that our proposed scheme can significantly improve the performance of the secondary system by comparing it with the other two baseline schemes.