We consider a cognitive radio system, in which a secondary transmitter harvests energy from a primary transmitter's RF signal. The secondary transmitter, which provides decode-and-forward relaying service for the primary system, transmits its own data by using downlink non-orthogonal multiple access (NOMA). A time-switching protocol is used by the secondary transmitter to harvest energy and decode the primary transmitter's information. Our objective is to achieve maximal secondary throughput, by optimally selecting the time portion used for energy harvesting and the secondary transmitter's power allocation in NOMA transmission. In this paper, two optimization problems are formulated, in which the secondary receiver performs or does not perform successive interference cancellation (SIC), respectively. Although the two problems are nonconvex, we devise a method to transform the problems into equivalent problems under different cases. Then, we theoretically prove that the objective functions of the equivalent problems are quasiconcave, based on which we develop two-level bisection search algorithms to solve the equivalent problems. Interestingly, we show that performing SIC at the secondary receiver does not always guarantee a higher secondary throughput than the case without performing SIC. Computer simulation demonstrates that our algorithm performs better than an equal power allocation algorithm and an orthogonal multiple access algorithm.