Purchase stucco caulk from a home goods store. If you cannot find stucco caulk, ask. Regular caulk does not work as well in this situation.
Cut the tip off of the tube with a utility knife about 1/2 inch from the tip. If cut much lower, the caulk could come out too fast and be difficult to insert into small areas of damaged stucco.
Insert the stucco caulk tube into a traditional caulking gun. Pull on the trigger until the material comes out of the gun freely.
Squirt a small amount of stucco caulk into the hole or small crack. The caulk should be level or just above the surface of the wall.
Smooth the caulk with a plastic putty knife and let it dry for 24 hours. Paint to match stucco if necessary.