My iCloud Email seems to be Offline and its mail is not moving into Apple Mail?

The mail application is corrupted , remove the accounts and once again add Add email accounts in Mail on Mac - Apple Support

If you are using secondary authentication account in which three answers for security questions , date of birth , rescue mail address and its password is added , and iCloud Keychain is set up in Mac , make sure you must have iCloud Keychain verification code in records as it will be required once you add gmail or whatever services used and iCloud account .

( a ) Open system preferences > iCloud > the box for keychain is checked , click on options , click on change security code .

( b ) Click on security code > enter new code in the fields and click on next , you have to enter the new code for the second time and click on next .

In the next window the country code is written , enter your trusted mobile number …….and click on done .

Again enter your Apple ID password to update your account settings and click on ok , hence the code is changed , so write down this code in a safe place .

Note : once the code is generated then remove mail accounts and once again add it , make sure must have correct credentials like Apple ID password , web mail address and its password in your records .

See the article Take email accounts offline or online in Mail on Mac - Apple Support

Take email accounts online

In the Mail app  on your Mac, do one of the following: 

• Take all accounts online: Choose Mailbox > Take All Accounts Online.
• Take a specific account online (if you have more than one account): Choose Mailbox > Online Status > Take [account] Online.

If it's an IMAP account all mails are saved on server , not a single mail will be lost , however syncing will take time using high network speed .

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