I am trying to add a power-over-ethernet adapter to my set-up. It has a small ground wire (yellow-green sleeve) with a small metal circle at its end going out of it which I assume I should connect somewhere. The power supply block does have a grounded plug, but the wire that goes to the PoE adapter only has 2 wires. What is the proper way to ground this appliance, and is it necessary to do so at all?
Your PoE is powering other devices, and these devices may or may not be bonded to earth.
No Ground Loop Please
To prevent a ground loop, all cabling should be grounded at most on one side, usually the router or PoE side.
In some low frequency applications, a ground loop can cause interference, but mostly ground loops are avoided to prevent equalizing currents that arise from different ground potentials at either end of the cable: the shield does not have sufficient gauge to function as a bonding conductor between electrical systems.
ESD Discharge please
To prevent electrostatic build-up in the shield of the cabling, the cabling must be grounded at least on one side.
Therefore, cabling with shielding has the shield grounded once at the PoE, and then not on the side of the powered devices.
If a device at a far end needs to be grounded, its ground must be isolated from the cable's shield, using isolated RJ connectors.
Supply a ground
The power adapter provided with the PoE does not supply a ground, and therefore the PoE needs to be supplied with a ground so that it can ground the cable shield through grounded and shielded female RJ connectors.
That yellow lead is the ground for the PoE, which will ground the chassis and the shields of connected ethernet wires.
Ground vs Neutral
Note also that the ground mentioned above is earth ground and it is not to be confused with the "0V ground" or neutral of the DC supply.
Data needs no Ground or Neutral
Also, the twisted wire pairs in the ethernet cable that carry the data use what's called differential signalling across both wires of the pair, and do not carry an earth ground or a 0V ground as reference.
The PoE and the ethernet data communication can work if you do not connect that ground lead.
In some applications (usu. not ethernet) the shield can pick up radio interference and couple it even into twisted pairs, and grounding the shield at one end creates an antenna making things worse.
That is likely a ground wire. The power brick is grounded, but it's only sending DC power to the POE device.
The only reason to ground it would be completeness. Ethernet itself does not need grounding because ground could introduce more noise on the circuit.
If you had a ground in between the transmitter and receiver it would create a ground loop. This ground loop would be formed by the cable and the return path would be mains ground as shown below. Any magnetic fields that flow through the loop would create a current along the cable (and the rest of the loop). Even if you isolated the signal wires this would be a problem because of mutual inductance between wires (wires run along side each other can couple currents from one to the other). This would inject noise (and cause potential bit error and packet loss).
I wouldn't bother with it myself. Ethernet is low voltage to begin with and that device is purely an isolated DC element anyways.