In this paper, a cognitive radio network is considered in which the secondary network (SN) consists of a source, a buffer-aided full-duplex decode-and-forward relay, and a destination, underlaid over a primary network (PN). An imperfect self-interference (SI) cancellation is assumed at the secondary relay (SR), such that the SI power is proportional to the transmit power of the SR. For the SN with limited power expenditure, a novel joint mode selection and power allocation policy is proposed to maximize the secondary throughput under the constraints of secondary power consumption and a limited average induced interference power at the primary destination. For delay sensitive SN applications, a statistical delay constraint is imposed in which the queue length at the SR can only exceed a specified threshold with a limited probability. In the two proposed policies, the SN decides optimally when to operate in half duplex mode and/or in full duplex mode, and be silent. To avoid data loss in the SN, buffer is used at the SR for data storage. Simulation results show that, for a given interference threshold and statistical delay constraint, the proposed policy outperforms the non-buffer full-duplex, buffer-aided half-duplex, and non-buffer half-duplex policies in terms of the average secondary throughput, the average secondary delay, and the secondary power expenditure.