Duty-cycling is an appealing solution for energy savings in densely deployed, energy-constrained wireless sensor networks (WSNs). Indeed, several applications, such as intruder detection and tracking, require the design of k-covered WSNs, which are densely in nature and where each location in a monitored field is covered (or sensed) by at least k active sensors. With duty-cycling, sensors can be turned on or off according to a scheduling protocol, thus reducing the number of active sensors required to k-cover a field and helping all sensors deplete their energy slowly and uniformly. In this paper, we propose a duty-cycling framework, called clustered randomized m-connected k-coverage (CRACC(mk)), for k-coverage of a sensor field. We present two protocols using CRACC(mk), namely T-CRACC(mk) and D-CRACC(mk), which differ by their degree of granularity of network clustering. We prove that the CRACC(m)k protocols are minimum energy m-connected k-coverage protocols in that each deploys a minimum number of active sensors to k-cover a sensor field and that k-coverage implies m-connectivity between all active sensors, with m being larger than k. We enhance the practicality of the CRACC(mk) protocols by relaxing some widely used assumptions for k-coverage. Simulation results show that the CRACC(mk) protocols outperform existing k-coverage protocols for WSNs.