Is it better to air dry or cover up a cut or broken skin so that it will heal faster?

Cover up. Dry wound is a dead wound. Moist wound healing is the current standard for healing wounds.
Cover! Studies show the best healing time is when the cut is kept moist. Use an antibiotic ointment and cover with a bandage. Change this twice/day with gentle cleaning to be sure it isn't getting infected. If redness, pain, or discharge increase, see your doctor.