My Mac Pro will not download mail though the icon is lit.
My Mail icon is lit, but mail will not download.
MacBook Pro, OS X Mavericks (10.9.5), null
Have you tried everything here?
https://support.apple.com/en-us/HT204075
Can you send mail from this account on another device?
Who is the mail provider? ( what is after the @ sign in the email address)