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Wi-Fi is one of the key wireless technologies for the Internet of things (IoT) owing to its ubiquity. Low-power operation of commercial Wi-Fi enabled IoT modules (typically powered by replaceable batteries) is critical in order to achieve a long battery life, while maintaining connectivity, and thereby reduce the cost and frequency of maintenance. In this work, we focus on commonly used sparse periodic uplink traffic scenario in IoT. Through extensive experiments with a state-of-the-art Wi-Fi enabled IoT module (Texas Instruments SimpleLink CC3235SF), we study the performance of the power save mechanism (PSM) in the IEEE 802.11 standard and show that the battery life of the module is limited, while running thin uplink traffic, to ~30% of its battery life on an idle connection, even when utilizing IEEE 802.11 PSM. Focusing on sparse uplink traffic, a prominent traffic scenario for IoT (e.g., periodic measurements, keep-alive mechanisms, etc.), we design a simulation framework for single-user sparse uplink traffic on ns-3, and develop a detailed and platform-agnostic accurate power consumption model within the framework and calibrate it to CC3235SF. Subsequently, we present five potential power optimization strategies (including standard IEEE 802.11 PSM) and analyze, with simulation results, the sensitivity of power consumption to specific network characteristics (e.g., round-trip time (RTT) and relative timing between TCP segment transmissions and beacon receptions) to present key insights. Finally, we propose a standard-compliant client-side cross-layer power saving optimization algorithm that can be implemented on client IoT modules. We show that the proposed optimization algorithm extends battery life by 24%, 26%, and 31% on average for sparse TCP uplink traffic with 5 TCP segments per second for networks with constant RTT values of 25 ms, 10 ms, and 5 ms, respectively.