In overlay networks, the cognitive sender () assists the primary transmitter () by broadcasting the superposed signal composed of both cognitive and primary information with a higher priority for the primary information to ameliorate spectral efficiency. Such different information priorities also facilitate efficient successive interference cancellation (SIC) at corresponding receivers for better system performance. To further benefit beside licensed spectrum accessing permission, we assume transmission of solely with energy scavenged from . Practically, energy scavenger has a nonlinear characteristic and circuit components almost suffer hardware imperfections. Further, path loss, shadowing, and fading are all present in practical wireless channels, which affect not only the scavenged energy but also the system performance. Consequently, this paper proposes a framework to analyze the performance metrics— outage probability and throughput—of overlay networks under realistic scenarios subject to hardware imperfections, energy scavenging nonlinearity, SIC-based signal detection, and versatile-and-general shadowed fading. This framework facilitates the system performance evaluation-and-comparison in essential specifications and serves well as a design instruction. Obtained results reveal that hardware imperfections affect the system performance more direly than energy scavenging nonlinearity does and the primary transmission performs considerably better than the secondary transmission irrespective of channel severities. Furthermore, the system performance can be adjusted and optimized versatilely by a multi-parameter set.