I'm going to give a bit of background to see if that makes a difference.
I have an icloud account that I literally use to sign in to itunes and I can't work out how to merge/change it or anything so that's how it stays - this is the primary. I don't use it for emails or anything.
I have an old personal icloud email that is still working but is mainly junk which is why I set up a new one so I could just abandon it. This one is the one that is working.
The new personal email - another icloud address - is the one that isn't working.
As far as I know - the password is connected to the app (Airmail) rather than the email account? And I so far haven't found anywhere to actually provide that password that has been created. But I'm assuming that because airmail is working with other accounts including an icloud one, that isn't the problem?