Q: Apple Mail - My mail account is recently been asking me for password and it will not allow me to logon - email works fine - the problem is with Apple Mail - any help?

There's a sequence of steps in the support article and linked pages for each of several possible scenarios of a connection failure in Mail. Please describe in detail what happened when you followed the applicable sequence all the way to the very end, without skipping any of the steps. That's the starting point for any further efforts to solve the problem.

Delete the password from Keychain Access (Applications/Utilities). While there, Keychain Access menu First Aid.

Keychain Issues - Resolve  see post by Kappy