In this paper, we investigate a wireless-powered relays assisted batteryless IoT network based on the non-linear energy harvesting model, where there exists an energy service provider constituted by the hybrid access point (HAP) and an IoT service provider constituted by multiple clusters. The HAP provides energy signals to the batteryless devices for information backscattering and the wireless-powered relays for energy harvesting. The relays are deployed to assist the batteryless devices with the information transmission to the HAP by using the harvested energy. To model the energy interactions between the energy service provider and IoT service provider, we propose a Stackelberg game based framework. We aim to maximize the respective utility values of the two providers. Since the utility maximization problem of the IoT service provider is non-convex, we employ the fractional programming theory and propose a block coordinate descent (BCD) based algorithm with successive convex approximation (SCA) and semi-definite relaxation (SDR) techniques to solve it. Numerical simulation results confirm that compared to the benchmark schemes, our proposed scheme can achieve larger utility values for both the energy service provider and IoT service provider.