Sounds like Facial Cysts with Ingrown Hair.
I have seen less cratering if the cyst it cut and the roots of the hair scraped out before it makes a big puss head that causes a scater from the skin rot up to the surface.
If it is regular cyst, you must be sure the skin enveloped cyst is scraped out to prevent it from reforming again. The bag that holds the cyst must be removed from under the flesh.