This paper investigates the time scheduling for a backscatter-aided radio-frequency-powered cognitive radio network, where multiple secondary transmitters transmit data to the same secondary gateway in the backscatter mode and the harvest-then-transmit mode. With many secondary transmitters connected to the network, the total transmission demand of the secondary transmitters may frequently exceed the transmission capacity of the secondary network. As such, the secondary gateway is more likely to assign the time resource, i.e., the backscattering time in the backscatter mode and the transmission time in the harvest-then-transmit mode, to the secondary transmitters with higher transmission valuations. Therefore, according to a variety of demand requirements from secondary transmitters, we design two auction-based time scheduling mechanisms for the time resource assignment. In the auctions, the secondary gateway acts as the seller as well as the auctioneer, and the secondary transmitters act as the buyers to bid for the time resource. We design the winner determination, the time scheduling, and the pricing schemes for both the proposed auction-based mechanisms. Furthermore, the economic properties, such as individual rationality and truthfulness, and the computational efficiency of our proposed mechanisms are analytically evaluated. The simulation results demonstrate the effectiveness of our proposed mechanisms.