Hello nosno, are you still having this issue? I found on google's site they received many reports about authentication but now resolved.
This link may be helpful to troubleshoot and identify if there are any issues reported with google services:
I'm having the same problem. I removed the mail account, and now it will not let me readd it. It says I already have a Google email account. I tried to remove the whole Google account, and it says that the app is integrated into the phone and that I will need to initialize (factory reset) the phone to remove the Google account. What now??
Authentication errors are usually resolved by removing your Gmail account. You must do a factory reset in order to remove the original Gmail account on your phone. Please do the factory reset and keep us posted.
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