Mail does not work

Please read If you can’t send or receive email on your Mac - Apple Support. This might also help: Mail Settings Lookup - Apple Support.

Here are some additional troubleshooting steps to follow.

• Temporarily stop using an email account:  (Apple menu) > System Preferences... > Internet Accounts > select your email service, and de-select Mail. Wait a moment or two, then select it again. Repeat for each email account you use, testing after each one.
• If all else fails, temporarily remove that Mail Account from that System Preferences pane, and then add it again.

Those steps are described in greater detail here: Set up Mail with email accounts - Apple Support.

Needless to say, if you are using—or have used at any time in the past—any non-Apple "anti-virus", "cleaning", or "Internet security" junk, anything at all in that broad category of useless garbage, don't. A lot of things won't work if you do.

Seeing that screenshot, Reinstall Safari (and to the Correct Version):

To get Safari to operate correctly, you must first delete Safari from your Mac, and then either:

• restore Safari from a backup; or 
• reinstall the macOS

For Note:

If you reinstall the macOS, nothing would be deleted. All intended, original files and Apps that came on your macOS would be reinstalled. It’s all about keeping your system files intact. 

Time Machine backup:

Important: Create a Time Machine backup of your Mac, so that you can have something to restore your Mac from, should anything go wrong.

To post a reply, go to the discussion in Apple Support C

Also, Try Clearing Your Temporary Internet Files: History and Cache

Cleaning your history would rid of all login credentials. Clearing the cache would result in all sites to loading like new and to a clear up-to-date default appearance.

alddrd Said:

"Mail does not work: Just updated to Mojave on Macbook Air 2014."

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Restart Your Mac:

Having just updated your Mac, restart the Mac. Things may still have yet to be be fully reconfigured, in accordance with the macOS Version's settings.

Having the same problem. Restarting several times does not help. Removing the accounts was possible, but now one cannot add the accounts as Google accounts anymore as it hangs in the "authentication"-pop up to Safari ... it opens browser but never gets or opens the page to enter password. This renders mail useless ...

App Specific Passwords: Gmail

I would assume you had no Application Key. Rather than using your Gmail login password, an Application Key is needed.

Why Application Keys are Used: Just think about it: If anyone knew your password, they would get all that info. The same goes for Hotmail.

Go Here: Sign in Using App Passwords - Gmail Help - Google Support

none of the above seems to help ... not even getting beyond the “authentication” pop-up, see screenshot. It does not open Safari, and not the login/ authentication page for Gmail