Wireless sensor networks (WSNs) involve more mobile elements with their widespread development in industries. Exploiting mobility present in WSNs for data collection can effectively improve the network performance. However, when the sink (i.e., data collector) path is fixed and the movement is uncontrollable, existing schemes fail to guarantee delay requirements while achieving high energy efficiency. This paper proposes a delay-aware energy-efficient routing algorithm for WSNs with a path-fixed mobile sink, named DERM, which can strike a desirable balance between the delivery latency and energy conservation. We characterize the object of DERM as realizing the energy-optimal anycast to time-varying destination regions, and introduce a location-based forwarding technique tailored for this problem. To reduce the control overhead, a lightweight sink location calibration method is devised, which cooperates with the rough estimation based on the mobility pattern to determine the sink location. We also design a fault-tolerant mechanism called track routing to tackle location errors for ensuring reliable and on-time data delivery. We comprehensively evaluate DERM by comparing it with two canonical routing schemes and a baseline solution presented in this work. Extensive evaluation results demonstrate that DERM can provide considerable energy savings while meeting the delay constraint and maintaining a high delivery ratio.