A doctor can tell if your child has hand-foot-and-mouth disease by the symptoms you describe and by looking at the sores and blisters.
Roast the beets for 50 to 60 minutes. Cook the beets until tender, checking them by inserting a fork into the center of each beet. When the fork glides in easily, the beets are tender enough to remove from the oven.
Preheat oven to 375 degrees. Toss beets with oil and a large pinch of salt. Wrap tightly in a parchment-lined piece of foil, and place on a baking sheet. Roast until tender, 45 minutes to 1 hour. Let cool. Rub beets with paper towels to remove skins. Puree in a food processor (you should have 1/2 cup puree).