Energy-harvesting sensor nodes (EHSNs) have stringent low-energy consumption requirements, but they need to concurrently execute several types of tasks (processing, sensing, actuation, etc). Furthermore, no accurate models exist to predict the energy harvesting income in order to adapt at run-time the executing set of prioritized tasks. In this paper, we propose a novel power-aware task scheduler for EHSNs, namely, HOLLOWS: Head-of-Line Low-Overhead Wide-priority Service. HOLLOWS uses an energy-constrained prioritized queue model to describe the residence time of tasks entering the system and dynamically selects the set of tasks to execute, according to system accuracy requirements and expected energy. Moreover, HOLLOWS includes a new energy harvesting prediction algorithm, i.e., Weather-Conditioned Moving Average (WCMA), which we have developed to estimate the solar panel energy income. We have tested HOLLOWS using the real-life working conditions of Shimmer, a sensor node for structural health monitoring. Our results indicate that HOLLOWS accurately predicts the energy available in Shimmer to guarantee a certain damage monitoring quality for long-term autonomous scenarios. Also, HOLLOWS is able to adjust the use of the incoming energy harvesting to achieve high accuracy for rapid event damage assessment (after earthquakes, fires, etc.).