Question: [![enter image description here](https://i.stack.imgur.com/I8qAm.png)](https://i.stack.imgur.com/I8qAm.png)I want to have a wired Internet connection before or after a PoE access point.

Is that possible?

Answer: 
For PoE to work, the last device before the PD (powered device, in your case an access point) needs to be the PSE (power sourcing equipment, in your case the PoE "injector", also called a Midspan). Your options are a) remove the PoE injector and install a switch which supports PoE instead between the firewall and AP, or b) run a second network cable from the firewall to a new switch in parallel with the AP cabling.
As far as I understand your problem: on the location of the AP you want another, wired Internet access? You can run the AP's patch cable through a switch to add ports to the network.

However, (usually) PoE doesn't pass the switch, so you'll need to connect the switch to the LAN side of the PoE injector. Then, from the switch, run another cable through the PoE injector and then to the AP as before.