Doctors say I have mastitis. All they wanna do is lance. Will I still have problem later?

No. Mastitis is when your breasts are inflammed/infected. In severe cases, antibiotics are given. If the process continues, you can develop an abscess. This is when pus present. If a large area is effected, an incision and drainage procedure is performed ("lancing"). This will evacuate the pus but you will now have a wound cavity to heal. Usually, this process does not recur in the same area.