Due to the ever-expanding applications of the Internet-of-Things (IoT), designing energy- and spectrally-efficient transmission schemes to support massive connections and devices is inevitable and still challenging. Thus, energy-harvesting (EH) and cognitive-radio (CR) systems are becoming more inseparable for future IoT networks. This paper analyzes the performance of EH-CR-IoT networks, where closed-form expressions for network metrics, such as GoodPut, collision probability and average packet delay are derived. In addition to the interference caused by spectrum sensing errors, our analysis also incorporates the primary user (PU) return interference into the different network metrics. Furthermore, the effect of primary network traffic behavior and IoT network parameters are investigated. To account for delay-sensitive packets, the average end-to-end delay of packets as well as delay violation probability in the IoT network are mathematically formulated and analyzed as quality-of-service (QoS) measures for network stability. Moreover, the derived metrics can be utilized to optimize the Goodput, subject to various practical constraints. Simulations are also performed to verify the theoretical results. Above all, the effect of energy-harvesting rate on GoodPut and IoT network stability is explored, which provides insights into determining the physical structure of the energy-harvesting system.