Thank you for your query.
I do not think that the spots on the face has to do anythng with the hand, foot and mouth disease. Hand, foot and mouth disease is associated with fever
and with ulcers in the pharynx, mouth and tongue which is not in your child's case.
Red rounded spots can occur because of various reasons and at times nothing needs to be done. I suggest that you can wait and watch. In case they increase in size or are associated with other symptoms then appropriate steps can be initiated after consulting the doctor.
I hope I have answered your query to your satisfaction.