Energy efficiency is a significant challenge for modern wireless networking devices. It is crucial for the Internet of Things devices, is required for battery-supplied user devices such as smartphones, and is advisable for high-performance devices such as wireless VR headsets. This paper examines the ability of modern Wi-Fi devices to achieve extremely low power consumption when they rarely send and receive data. Two recently developed mechanisms, namely Target Wake Time (TWT) and Wake-Up Radio (WUR), are studied. The first one allows stations to schedule the frame exchanges in advance, while the second one introduces a low-power radio for control information exchange. Although TWT and WUR differ significantly, they both suffer from the clock drift effect that significantly degrades their performance in the case of rare traffic. The paper describes these mechanisms, focusing on their revolutionary features, and presents mathematical models to evaluate the impact of this effect on TWT and WUR mechanism efficiency in terms of energy and channel time consumption. The paper also proposes and thoroughly examines various approaches to the joint and separate usage of TWT and WUR in Wi-Fi networks.