The irregular black spots are likely pecking injuries common to all chickens as they attempt to settle disagreements. The white may be simply dryness, or it could be the beginning of a fungus called favus. To test whether it's just dryness, smooth on a little lubricating cream, Vaseline, or lotion. If it disappears, it's just dry skin.
If the white remains unchanged after lubrication, it may be a fungus. If you have Blu-kote, that will kill the fungus, or use athlete's foot cream or vaginal yeast cream until it goes away.