Repair or restore monitor screen coating

While trying to “buff out” a scratch on a monitor screen with an oiled cloth, I inadvertently removed some of the anti-glare coating. The spot is in the center of the screen, about the size of a dollar coin. Although the monitor still works fine, that spot is annoying. Does anyone know of a remedy for this? Is there any product that can re-coat the screen after-the-fact? Even if it’s not perfect, any improvement would be helpful.