Filling those holes will be a 4-step process
1. Make 9/16" long wood plugs if the MDF is 3/4" thick, the plugs can be cut off dowels. The plugs can be longer but you will have to cut them off on the inside.
2. Glue in the plugs, make sure they are at least 3/16" below the finished surface. Why? MDF expands and contracts equally in all directions with changes in humidity. A dowel is long grain and does not move at all. If the dowel was cut and sanded flush with the surface a circular shadow would eventually appear. Let dry overnight.
3. Fill the shallow hole with Bondo or polyester glazing putty. Try the polyester glazing putty. Available at Ace Hardware.
4. Sand smooth, fill and sand again, you might need 3 applications to get a smooth patch feathered into the surrounding area. MDF is soft, don't sand a trench around each hole, wrap sandpaper around a wood block.