This is a tough one. In most cases a charging port is usually soldered onto a circuit board, so one that gets pushed inside means it has come off the printed circuit and would need to be soldered back on to restore the electrical connections. If that was the case, then just gluing it back in place wouldn't make it work again, but the problem here is that we just don't know.
There is a chance the port is connected via something like a flex cable or wires, in which case it might still be capable of working when pushed out of place or still working, but I've been unable to find any information on that port, not even a photo of a motherboard once it's been removed from the case. System76 has a picture of the laptop with the bottom cover removed, but unfortunately there are covers over the connectors on the left side of the laptop, obscuring the view of the charging port.
[image|3689630]
System76 does have a web page showing how to do some minor repairs on the computer, but it stops far short of removing the motherboard. However, it will show you the right way to remove the bottom cover and other components as well.
[link|https://tech-docs.system76.com/models/pang15/repairs.html|Parts & Repairs - System76 Technical Documentation]
At this point about all we can do to help you is to suggest you open up the laptop, figure out how to remove the covers obscuring the charging port, and take some clear, high resolution photos so we can assist in figuring out how to fix the port.