Dear Emma Robertson,
Welcome to the Microsoft Community.
After reading, I have a general idea of your problem. I think your problem is related to the fact that basic Outlook authentication has been deprecated and this is the last public announcement from Microsoft.
I would like to know what your current version of iOS is. The system or application may be an older version that does not support accounts that use reauthentication.
Update your iOS version to the latest version and, using the latest version of Apple Mail, add your account following instructions below again and see if the issue is resolved.
On iPhone,
- Go to Settings > Email.
- Tap Accounts, then tap Add Account.
- In the Add Account dialog box, choose Outlook.com.
If removing and re-entering your account doesn't resolve the issue, do the following:
- Remove the account again.
- Go to Settings > Safari and tap Clear History and Website Data.
- Restart your iPhone.
- Add the account again and proceed with the account sign-in flow Outlook.com.
Of course, if you're still having trouble adding your Hotmail account in Apple Mail, you can also download the latest version of the Outlook app for iOS, or even use Outlook web.
Please feel free to contact me if there are any further updates on this issue, I am here to help you further if needed.
Best regards
Stan - MSFT | Microsoft Community Support Specialist