This paper focuses on enhancing the energy savings of EPON-based backhaul for 5G and beyond networks. Here, we introduce a novel technique of the desynchronization of wake-up cycles among Optical Network Units (ONUs) in Low-Power Mode (LPM). In this technique, the Optical Line Terminal (OLT) decides the wake-up slots of all ONUs such that they are spread out and staggered, maximizing the likelihood of fewer ONUs waking up simultaneously. We have shown that this technique can be implemented in addition to any of the existing protocols to improve the energy savings of the protocol. We formulate an optimization problem to achieve wake-up desynchronization while ensuring latency requirements. To address this problem, we present a polynomial-time 1-approximation algorithm. The convergence and complexity of the algorithm are analyzed. Through simulations, we illustrate a significant increase in energy savings and a reduction in the probability of delay violation by implementing our proposed method.