Seems to me that whatever you are using to "clean" is the cause of the residue in the photo. I have used Sprayway myself for windows and it is far from streak-free. You might get better results with a mild solution of distilled water and white vinegar.
For windows, I spray the window and then buff it dry with newsprint (yes, newsprint), but for the turntable cover I would use a soft, liny free cloth like one used to clean computer screens.
Not sure why all the talk about swirls, when that isn't the issue.