How are you treating your Betta for ich? If your Betta had/has ich then its possible the white specs are the ich once its fallen off the Betta. If the shelf has dust & the cup isn't sealed, which it shouldn't be, then dust is going to get into the water. I don't know what kind of normal set up your Betta is kept in but I wouldn't keep him that cup, if that is what you are doing, I would move to his normal, bigger, heated tank. I've only treated ich with the heat & water change method which has been greatly successful for me.