Was the sugar dissolved in water when you poured it in?
If so, you'll likely be fine. Also, the oxygen effects will be minimal to none provided you don't keep the bottles for years. Even so, you might not notice it.
If you didn't dissolve the sugar first, then yes it's possible that you'll have some inconsistent carbonation.
In Process - Russian Imperial Stout, Nelson Sauvin Rye IPA, Mild No.3
In Kegs - Barley Wine, Apfelwein, Wild BlackBerry Wheat, Coffee Oatmeal Porter
Gone - so many :(