Wipe the headboard with a damp microfiber cloth to remove any dust and dirt on the surface.
Mix a bowl of one part white vinegar and one part water, if the headboard is especially dirty. Dip a microfiber cloth in the solution, wring it out, then wipe the headboard down. Wipe the headboard again with a microfiber cloth moistened with water only and allow it to dry.
Fold a clean, dry microfiber cloth into a small square. Apply a small amount of ultra fine metal polish to the cloth.
Rub the scratched area of the bronze headboard with the polish in a tight circular motion. Do not apply hard pressure.
Buff the polished area with a clean, dry microfiber cloth. Repeat until the scratches fade.
Apply polish to the rest of the bronze headboard to give it an even luster, buffing the polish away as you go.
Moisten a fresh microfiber cloth with liquid wax. Apply the wax to the bronze headboard the same way you applied the polish, including buffing the areas as you go. This step isn't mandatory, but it will add a protective layer that will resist darkening and future scratches.