Energy harvesting (EH) delivers a unique technique for replenishing batteries in Internet of Things (IoT) devices. Equipped with an energy harvesting accessory, EH-enabled sensor nodes/IoT devices extract energy from ambient resources such as solar or radio frequency (RF) signals. Relying on residual battery or/and harvested energy, sensor nodes in an IoT network perform data exchange activities. Otherwise, the delivery of sensed data would be delayed until sufficient energy is harvested. In this paper, we propose an on-demand energy requesting (OER) mechanism for improving the delay performance of a wireless EH-powered IoT network. The proposed scheme acquires energy when necessary from an energy transmitter that is capable of transmitting energy via directed RF signals. Furthermore we develop two associated discrete time Markov chain (DTMC) models to analyze the performance of the OER scheme, targeting at a generic synchronous medium access control (MAC) protocol. Using the proposed DTMC models, we evaluate OER with respect to average packet delay, network throughput, packet loss probability, and packet reliability ratio by employing a specific synchronous MAC protocol. Numerical results obtained from both analysis and discrete-event simulations coincide with each other, indicating the accuracy of the models and revealing the behavior of EH based packet transmissions.