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Have you tried retinol cream? You could try low dose in a small area avoiding the white spots. One of the risks of creams that will work is the risk of bulls-eye spots as different areas lighten differently. But it's worth a try if the dermatologist won't help or is unaffordable.
Remember, whatever is tried do small areas at a time as no one knows how your skin will react. Also use the lowest dose to start.