This paper investigates the maximum secondary throughput for a rechargeable secondary user (SU) sharing the spectrum with a primary user (PU) plugged to a reliable power supply. The SU maintains a finite energy queue and harvests energy from natural resources and primary radio frequency (RF) transmissions. We propose a power allocation policy at the PU and analyze its effect on the throughput of both the PU and SU. Furthermore, we study the impact of the bursty arrivals at the PU on the energy harvested by the SU from RF transmissions. Moreover, we investigate the impact of the rate of energy harvesting from natural resources on the SU throughput. We assume fading channels and compute exact closed-form expressions for the energy harvested by the SU under fading. Results reveal that the proposed power allocation policy along with the implemented RF energy harvesting at the SU enhance the throughput of both primary and secondary links.