"How should one treat an infected wound besides washing it with mild soap and applying polysporin?"

To treat an infected wound besides washing it with mild soap and applying polysporin, you can also apply topical antibiotic ointment over the entire surface of the wound and cover it with a bandage that should be changed daily. Avoid using skin disinfectants during cleaning, only use water and mild soap. If the redness increases beyond what you have outlined, if the pain worsens, if you develop a fever or chills, or if there is no improvement by early next week, it is important to seek medical attention for a more aggressive treatment to eliminate the risk of infection.