What should I need to do about a minor infection from a cut on a 5-year-old?

Modern tx. Avoid using alcohol, peroxide, hibiclens, betadine, and bleach solutions. Inflammation ( red, hot, swollen, tender) is important to healing so anti-inflammatories (motrin, advil) are bad. Cover the area with a clean dressing changed every other day and protected from trauma. Antibiotics do not help healing unless it is truly infected. Ssoaks are an old fashioned treatment not used by wound doc.
Soak! Depends a little on where the cut is--hands and face get worse quickly. Wash the wound well, soak for about 20 min in clean warm water three or four times a day, and cover in between soaks, perhaps with some Bacitracin ointment. If worse or not better in a day, see your doctor. If there is any chance of a thorn or sliver in the wound, go now.