Mail app cannot connect to iCloud account

You have few iPhones in which the Apple ID is say example1@gmail.com and two Macs that has also the same Apple ID and password used in iPhones .

A second Mac ( that is causing trouble ) in which you were previously signed as example1@gmail.com , then you opened system preferences > iCloud did you signed out completely , and didn't entered the previous Apple ID address say example1@gmail.com and then tried to create a new Apple ID say example2@icloud.com

Note : are you using secondary authentication Apple ID account in which three security answers for respective questions , date of birth , a rescue mail address and a password is added or did you sign in Mac with an Apple ID account set up with two factor authentication .

That has to be cleared the few iPhones , a Mac that is working perfectly and another Mac that is effected is synced on the same network , what is the version of iPhones , are they updated / upgraded with latest version .

Just this I wanted to suggest now. Happy for you it worked. You could have started too by unchecking the button iCloud > Mail close and check the same button again.

Have a nice evening (or what ever around the globe)

marek

On thhe Mac where it is not working check the following:

Preferences > on Top "Apple ID" > iCloud to the left > is the radio button "Mail" checked?

Best greetings

marek