You can move a published page to a different category by using the 'Move' menu available on any page or page list view:

This will bring up the following menu allowing you to choose a different category and folder to move this page into.

You can also update a page category when editing a page. From within the edit view for a page use the button on the top left to choose a new category for the page (you can also create a new category from this view):

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