In this paper, we investigate the performance of a full-duplex (FD) Internet-of-Things (IoT) network with hardware impairments (HIs) over generalized α − μ fading channels, where IoT devices transfer their information to the access point via a simultaneous wireless information and power transfer (SWIPT) enabled relay node. The α − μ fading channel explores the non-linearity of the propagation medium and includes Nakagami-m, Rayleigh, exponential, one-sided Gaussian, and Weibull fading channels as special cases. For SWIPT enabling, we consider a power splitting (PS) receiver at the relay node. Further, we consider a practical scenario where the system suffers from residual self-interference (RSI) due to imperfect self-interference cancellation. We derive the closed-form expressions of outage probability (OP), asymptotic OP, system throughput, and energy efficiency over α − μ fading channels. The diversity order of the considered network at high SNR is obtained. An optimal value of the PS factor is obtained which depicts optimum outage performance. We highlight the impact of HIs, RSI, PS factor, and other system parameters on the performance of the considered network. Finally, Monte-Carlo simulations validate the derived closed-form expressions.