internet accounts on mac - Email Mismatch Error Messages

I am based in the UK and when Google email accounts first came to the UK they were called ‘Googlemail’ and I registered a Googlemail email account. Some years later, those of us in the UK with Googlemail accounts were able to now use the ‘Gmail’ email address instead of ‘Googlemail’. Hence I have been using Gmail as my email ever since. What appears to have happened is that Google then retained 2 email accounts (1) Googlemail; and (2) Gmail.

For some inexplicable reason now, when I try to access my Gmail account using Mail on my mac, I have to do the 2 factor authentication, and then it states ‘Email Mismatch Error’. It now also affects my icloud email account. I now cannot receive or send emails using Gmail or icloud. I can send and receive emails using Gmail and icloud when I use Firefox or Googlechrome.

Help!