Get GoDaddy to move you to a Linux server. You're on a Windows shared server, and that's why you see the index.php in permalinks. Windows requires that for pretty permalinks. Linux servers do not. And linux is more secure and less buggy anyway.
See https://support.godaddy.com/help/article/1285/switching-your-hosting-account-operating-system The change will be transparent to you and users.