After updating to macOS Mojave 10.14.4 two days ago and still, I have an issue with taking online Google accounts in Apple Mail.app.
This problem appeared immediately after the update, although it wasn't there before. Also after this update, Apple Mail requested to re-login through Safari and it was simple authentication.
Best Answer
A quick Google search reveals that it's a bug in Mail.app in macOS Mojave 10.14.4 update.
From this post on MacRumors, a popular independent Apple news blog:
Additionally,
Appears that the issue has been acknowledged and the fix is a work in progress.
Some other links worth checking which points to possible resolution:
The TL;DR on Apple’s macOS Mojave 10.14.4 Gmail/G-Suite Issue: It’s Fixed
Apple’s macOS Mojave 10.14.4 Update Includes Gmail Bug – We Have a Possible Fix
Some macOS Mojave 10.14.4 users having problems with Gmail & G Suite authentication in Apple Mail
Some macOS Mojave 10.14.4 users having problems with Gmail and G Suite authentication in Apple Mail