If you have filament stuck in the extruder on your Elegoo Centauri Carbon, don't panic and start pulling everything apart right away. This can happen from a bad unload, broken filament, a clog near the hotend, or a small piece of filament getting trapped inside the extruder.
The first thing I'd try is heating the nozzle to the normal printing temperature for the filament you're using. Once it's hot, use the unload function and gently help the filament back out. Don't yank on it too hard because if the filament snaps inside the extruder, you can turn a simple problem into a more annoying one.
If the Centauri Carbon won't unload the filament, try removing the PTFE tube from the top of the extruder so you can see what's happening. Sometimes you'll find that the filament has broken right around the entrance to the extruder. If you can grab the remaining piece with a small pair of pliers or tweezers, you may be able to remove it without taking much else apart.
If the filament is stuck farther down, you'll probably need to open up the extruder. The good news is that the Centauri Carbon extruder can be disassembled, so a broken piece of filament isn't the end of the world. Once you have access to the gears and filament path, look for a small chunk of filament wedged between the gears or stuck before it reaches the hotend.
Another possibility is that you're dealing with a nozzle or hotend clog rather than the extruder itself. If the extruder gears are turning or clicking but filament isn't coming through the nozzle, heat the hotend and try manually pushing filament through. If it still won't move, you'll want to start troubleshooting the nozzle and hotend.
I'd also check the end of the filament after you get it out. If it's badly deformed, mushroomed, or has a large blob on the end, that can give you a clue as to why it became stuck in the first place.
For a visual walkthrough, I'd recommend searching YouTube for “Elegoo Centauri Carbon extruder disassembly” or “Centauri Carbon filament stuck” before completely taking the toolhead apart. Seeing where the screws, gears, and connectors are located makes this job a lot easier than going into it blind.
Most Centauri Carbon filament jams aren't a major repair. Start with heating and unloading the filament, then check for broken filament around the extruder. I'd only move on to disassembling the extruder or hotend if those easier fixes don't work.