Aiming at finding the $k$ nodes with the highest readings among sensor nodes, top- $k$ query is a very important and useful query in wireless sensor networks (WSNs). Current studies about top- $k$ query in WSNs mainly focus on always-on WSNs (A-WSNs) where sensors always stay awake and little research pays attention to top- $k$ query in duty-cycled WSNs (D-WSNs) in which sensors are dynamically awake and asleep to conserve energy. In this paper, we originally analyze the research issues when implementing top- $k$ query in D-WSNs and then propose the DRC-WSNs [D-WSNs with data replication (DR) and connected $k_{c}$-neighborhood (CKN)] scheme for performing top- $k$ query. Extensive theoretical analysis and simulation results provide the following two new insights regarding top- $k$ query in D-WSNs: 1) Implementing top- $k$ query in DRC-WSNs can achieve the best tradeoff with respect to query data accessibility and query cost (total energy consumption and query response time), compared with implementing top- $k$ query in DC-WSNs (D-WSNs with only CKN) and A-WSNs; and 2) the query data accessibility and the query cost of DRC-WSNs, DC-WSNs, and A-WSNs can be tuned by varying the number of neighbor nodes that the data are replicated to during DR, the $k_{c}$ in CKN, and the total sensor nodes number.