icloud keeps asking password on mac

after turning on two-factor authentication from my iphone, mail on my mac (snow-leopard) constantly keeps asking for my (correct) password for icloud. I belong to those who have previously been on MobileMe & after 2012 have been migrated to icloud.

I have signed out & in again, as suggested, but to no avail. I have also deleted my icloud account from mail (preferences) but trying to add it again is met with the message: "logging in to the MobileMe IMAP has failed. make sure the user name & password are correct" etc.

how can I get back to my icloud on mail? it works fine on the browser & on my iphone 4S. please help!