The sound works, but only produces a low sound. Additionally, I have check the system preference's sound section, and checked the output, mute check, and made sure the headphones were chosen as the output. However still emits low sound. Is the jack damaged in some way and how can I fix it?
I'm assuming that you've had no issues when using your computer's speakers to produce output so you could download an EQ like Boom 2 or eqMac to bump up the volume via software. Just be careful using your speakers afterwards if you have to turn it up really high.
Another solution which I think is a bit safer but can be a bit of a hassle depending on your personal preference would be to buy an external sound card. Depending on what you get you may gain functionality and if it doesn't work you can return the item and you can be pretty sure it's a software issue.
I also would like to make sure that you're not using high impedence headphones that need an amp because if you are then that would explain your issue but I'm assuming you're not.
Make sure the cord is plugged all the way in and that there isn't anything in the audio jack. I've had this issue happen several times before where some lint or dust buildup wasn't letting the headphones plug all the way in.