Sorry to hear that you cut your face, but at the same time I am glad that you didn't sustain a more serious injury!
When you develop a cut like this under the facial hair, it may initially lead to some disruption in hair growth. However, as long as the cut is not serious and does not get infected, over a period of days to weeks the scab should fall off and the hair should grow back in. You should definitely avoid shaving over the cut, as this will irritate the new skin and could lead to a risk of infection, delayed healing, and a more prominent scar. You can also apply some antibacterial ointment or some vaseline to keep the area most and to promote healing.
This is an issue that you should talk to your primary care doctor
or your dermatologist
about, especially if you have any concerns about proper healing or scar formation. They will be able to evaluate the cut to make sure there are no signs of infection, which could significantly delay healing. If there is an infection, they can prescribe antibiotics to clear it up and to promote healing.