do you get mail on those accounts? and this pop up occurs when changing folders, or does not mail download?

I have 4 addresses for picking up mail from Gsuite email accounts to do with a volunteer organisation I work for but don't want to faff with horrendous Gsuite webmail, I also have 1 conventional sky address (this one). When I open TB I get the message even if looking at sky emails. It doesn't pop up when I move between inboxes but it pops up at various intervals but don't know what triggers it. I typed an email from one Gsuite a/c to my sky a/c and the popup came up while typing but when I did a second one it didn't popup. I typed one from another Gsuite a/c to my sky a/c and it didn't come up so I don't think it's 'first use' or anything like that, I suspect timing may be an issue. Could TB cause this?

OK it defienitely pops up whenever I open TB.

ok, so you are being asked to re authenticate with google using the page displayed. it is actually the first of a number of steps. Have you done that? Unless you do you will not receive any mail from gmail/google as the account is not authenticated.