I think this may not be an actual problem for you. It's possible that the duck wasn't able to keep herself clean, possibly due to insufficient clean water, and dirt and gunk built up around her eyes. This usually happens toward the front of the face. Because the dirt mixes with eye gunk (sorry for the technical language!), It gets very solid, and when the duck tries to clean it some feathers around her eye may tear out, and it looks like that has happened.
If you decide to let the duck handle things on her own, I recommend a low three-gallon tub, plastic or rubber, that they sell in most farm and feed stores. Fill with water, place it in her pen, and let her wash on her own. Change the water when it gets dirty and make sure she has straw, not mud, to stand on when she gets out of the tub. It may take a couple of days for her to get it all out.
For these ducks, it's summer, and they are in the open barnyard, not their pens, but this is the tub I am talking about:
Alternatively, this could probably be a good moment for bonding with your new duck. If you have a dishpan that fits inside your sink or bathtub (
Walmart has them for $1.59, if you don't), fill it with clean warm water. Have a cup and a washcloth nearby. Your duck will want to flap wings in the water, but don't let it. Hold the wings close to the body and lower the duck into the water. Use the cup to pour a warm water over the duck's back.
Once the duck realizes that this is duck spa time, she will probably relax. Dip the washcloth into the water and wipe the back of her head first, then dip the washcloth into the water and showly drip it onto the bump near her eye. You can scratch at the dirt very gently with your fingernail, if she lets you. That will gradually help to loosen the dirt. I have done this for ducks who were going to shows, and it takes a long time to get the dirt out without removing feathers with it.
Full disclosure: dress to get wet.