I hope you can help me. I just upgraded to MacOS Sierra. I get the "Unable to verify account name or password." I'm pretty sure I have the correct password. I read that deleting and adding back the account seems to work BUT i already have a lot of messages in my inbox that I don't want to risk deleting. Thanks in advance for your help!
I had this - I found that if you put your password into the little dialogues that Mail pops up, it always fails. If you instead go into Mail settings and enter the password in the server settings tab for the account and save, it works.
Incidentally I have yet to see a config that works correctly with "Automatically manage connection settings" enabled.
If you enable two-factor authentication, Mail App does not support it for POP accounts. To solve this: go into your google account (google.com > on the top right click on your icon, My Account), then select "Sign-in & security" and in there you can create an "App Password". Once you have a unique App Password for Mail, go into Mail/Preferences/Accounts. Select the POP account you want and the "Serve Settings" tab. In there enter your password. For the secret step: Quit Mail App and relaunch it. That is what worked for me.