It appears that you have traditional plaster and, for what ever reason, the plaster finish coat is breaking away from the scratch coat. The paint itself is holding fine.

I would square off the plaster to where it is still firm , and then patch it. I like to use "hot mud" for such patches. Hot mud is like drywall compound, but it dries chemically. It comes as a dry powder and is mixed with water. It is available in several set times: 5, 20, 45, etc. It sands fairly easily. It is a good practice to dampen the scratch coat of plaster to help assure adhesion to it. After patching, your walls do have a texture which will have to be matched. There are spray cans available for such small patches. Then, prime and paint.