Low harvested energy poses a significant challenge to sustaining continuous communication in energy harvesting (EH)-powered wireless sensor networks. This is mainly due to intermittent and limited power availability from radio frequency signals. In this paper, we introduce a novel energy-aware resource allocation problem aimed at enabling the asynchronous accumulate-then-transmit protocol, offering an alternative to the extensively studied harvest-then-transmit approach. Specifically, we jointly optimize power allocation and time fraction dedicated to EH to maximize the average long-term system throughput, accounting for both data and energy queue lengths. By leveraging inner approximation and network utility maximization techniques, we develop a simple yet efficient iterative algorithm that guarantees at least a local optimum and achieves long-term utility improvement. Numerical results highlight the proposed approach’s effectiveness in terms of both queue length and sustained system throughput.