For PoE to work, both ends of the Ethernet link carrying power need to support the same flavor of PoE. I've never seen a USB Ethernet adapter that supports receiving (acting as the "powered device" (PD), not the power supply for) any flavor of PoE.
I doubt a PoE-receiving USB Ethernet adapter exists, because PoE is thought of as a way to power whole remote embedded devices, not just their NICs, and I don't think there's a commonly implemented way for a USB device to power a USB host (not until USB type-C, that is).
So you won't be able to make this work until/unless you find a USB Ethernet dongle that is designed to act as a PoE receiver for the same flavor of PoE as your PoE-supplying hub/switch/midspan power supply (power injector/splitter) supplies.