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On my Facebook profile, I accidentally put my mobile number in the section for my home number.
When I try to move it to the mobile section I get an error: "This phone number already exists in your account."
How can I delete the home number, or move it to the mobile section?
It does not seem this is possible, you reach a "road block" when going to the url as a Facebook Page (instead of a user), with the following message,
You are using Facebook as [Name of Facebook Page]
To access this page, you'll need to switch from using Facebook as your page to using Facebook as yourself.
The setting you're looking for is actually on the Privacy tab in your settings.
Change it from "Everyone" to "Friends", and only people who you are in your Friends list will be able to find you by your phone number.
Best Answer
You do it in two steps:
The thing is to save the edit between deleting it from one section and adding it to another one.
(I just tested it and it works.)