It looks as if you may have left the wheel cleaner on the wheels too long before rinsing it off and it’s dried on and stained or mildly etched the lacquered surface.
I find with regular washing (every week or couple of weeks) with a good quality ph neutral car shampoo, there’s very little need to use a dedicated wheel cleaner - it does help of your wheels are well protected with a sealant as this helps to make them easy to clean).
If you do feel the need to use a dedicated wheel cleaner, Valet Pro Bilberry wheel cleaner is ph neutral and I’ve used it on diamond cut alloys with no adverse effects to the wheels. Again, don’t leave it on to long, otherwise your wheels may be stained pink!
https://www.cleanyourcar.co.uk/valet-pro-bilberry-wheel-cleanerThe video at the link posted above shows it in action.
Going back to your current problem; I’d try polishing the affected areas gently with a very mild polish that contains very little by way of abrasives, followed up with a couple of coats of wax or sealant to protect the surface against the winter elements.