Many connected devices are expected to be deployed during the next few years. Energy harvesting appears to be a good solution to power these devices but is not a reliable power source due to the time-varying nature of most energy sources. It is possible to harvest energy from multiple energy sources to tackle this problem, thus increasing the amount and the consistency of harvested energy. Additionally, a power management system can be implemented to compute how much energy can be consumed and to allocate this energy to multiple tasks, thus adapting the device quality of service to its energy capabilities. The goal is to maximize the amount of measured and transmitted data while avoiding power failures as much as possible. For this purpose, an industrial sensor node platform was extended with a multi-source energy-harvesting circuit and programmed with a novel energy-allocation system for multi-task devices. In this paper, a multi-source energy-harvesting LoRaWAN node is proposed and optimal energy allocation is proposed when the node runs different sensing tasks. The presented hardware platform was built with off-the-shelf components, and the proposed power management system was implemented on this platform. An experimental validation on a real LoRaWAN network shows that a gain of 51% transmitted messages and 62% executed sensing tasks can be achieved with the multi-source energy-harvesting and power-management system, compared to a single-source system.