To power the Muon over PoE I need a PoE HAT and move the jumper on the board to connect the 5V_IN and center pin. All clear.
My question is:
Will the battery be charged when powered by PoE and will it switch automatically to battery power when the PoE power is lost?
The Muon will charge the battery from PoE when the jumper is set correctly. The Muon will switch to battery if PoE power is lost, as well.
The HAT 3V3 will be powered from the Muon, but HAT 5V won't be powered, since that's configured as a power input, not output, when using PoE.
Great, thank you for the clarification!
Time to order the PoE HAT 
Make sure you get one designed for the Raspberry Pi 5.
The Pi 4 PoE HAT connectors are in a different position and don't fit the Muon.
Yes I will go with the Waveshare one mentioned in the Muon datasheet.
I just received the Waveshare PoE hat (G) so I changed the jumper to connect 5V_IN and the center pin and hooked it up to my PoE router. No success...
The red power LED on the PoE hat lights up and then dims down during 2 seconds and then lights up again which keeps repeating. I measured the 5V on pin 2 and the voltage shows the same pattern as the power LED intensity. It shoots up to 5V and then drops off. None of the led on the Muon are lighting up.
I tried PoE and PoE+ ports but with the same result.
According to the router nothing is connected to the ethernet port
I power a Home Assistant Yellow over PoE from the same router without problem.
Do you think it's a communication problem between the router and PoE hat, a defective unit or is there anything to troubleshoot? Any advice is welcome!
Edit: someone on Amazon wrote that he had a similar problem when connecting a Pi 5 to his Unifi device. I also use Unifi.
The Waveshare PoE HAT was supposed to work, but I didn't try it myself before. However, I just tried it now, and how did you get it to fit? There's a large component labeled 4R7 on the bottom of the HAT that hits the M-SoM that prevents it from sitting in the proper position. Some other components might interfere as well.
OK, the H version includes the header extensions, I'll order one of those.
Tomorrow I can get my hands on this passive PoE adapter.
Would that be worth a try?
https://www2.mouser.com/ProductDetail/Phihong/POE61W-560DG-R?qs=RF%252Bx3zajg8JdU%252BRpvEpy8Q%3D%3D&srsltid=AfmBOor6LYHu9_JjPxc63mx8ZPuFAyVlvlq_JsJcw4L8mkn-WTvF5JDA
Ok, I found the problem:
The Unifi PoE is pretty picky and does not supply power when the power draw is too small for its taste. The Muon alone doesn't draw enough power to get above this limit.
I added an additional load by connection a bunch of white LEDs to the 5V pin and everything started as expected. I'm not sure what the cutoff value is but the Unifi app is reporting 3.2 - 3.5 Watt at the moment.
Thanks @rickkas7 for looking into it and hopefully other people with the same issue will find this topic.
That's interesting! There can be a minimum load with PoE but I expected that it would only affect sleep mode, not normal operations.
Indeed. I removed some LEDs and it's running stable at ~ 1.8W.
I will try to find the limit later but I assume every brand (or model) will have a different limit.
