The use of mobile sinks (MSs) can address several challenges for traditional wireless sensor networks (WSNs), such as energy holes, meanwhile it also introduces some new issues. Especially, for the path constraint MS, the fixed trajectory and uncontrollable moving speed results in significant delay and even buffer overflows. Existing research mainly focused on delay-tolerant applications which are unsuitable for delay-sensitive data or adopted the greedy policy to collect data which is inefficient in energy. To minimize the energy consumption and guarantee delay requirement simultaneously, we propose a delay-sensitive and energy-efficient routing scheme. Firstly, to obtain the MS position, we propose a K-nearest neighbor-based prediction model, which can reduce the energy and delay overhead of the location update. Based on the prediction result, an efficient cluster routing scheme is designed. To ensure both search speed and the optimality of the cluster head (CH), we improve the Reed-Deer algorithm (RDA) as IRDA, and use the IRDA to find out CHs. To remove the redundancy of the CH, a removal method is proposed. After constructing clusters, a method is used to determine the next-hop of the CH by using the predication position of the MS, the received timestamp and the delay requirement of the data. A fault tolerance mechanism is supplied to deal the predication error and other unexpected situations. By applying IRDA and the prediction model, the proposed routing scheme avoids the frequent construction of new routes, reduces the hops of the end-to-end path, minimizes network energy consumption, and ensures timely data transmission. The comparison results indicates that the proposed protocol can effectively improve the network performance while guarantee delay requirement. Especially in realistic scenario, the proposed algorithm can effectively reduce energy consumption and the end-to-end delay and improve the packet delivery ratio.