Energy harvesting techniques become increasingly popular as power supplies for embedded systems. However, the harvested energy is intrinsically unstable. Thus, the program execution may be interrupted frequently. Although the development of non-volatile processors (NVP) can save and restore execution states, both hardware and software challenges exist for energy harvesting powered embedded systems. On the hardware side, existing power detector only signals the ``poor'' quality of the harvested power based on a preset threshold voltage. The inappropriate setting of this threshold will make the NVP system suffer from either unnecessary checkpointings or checkpointing failures. On the software side, not all tasks can be checkpointed. Once the power is off, these tasks will have to restart from the beginning. In this paper, a task scheduler is proposed to maximize task progress by prioritizing uncheckpointable tasks if power is weak so that they can finish before the power failure. To assist task scheduling, three additional modules including voltage monitor, checkpointing handler, and routine handler, are proposed. Experimental results show increased overall task progress and reduced energy consumption.