Thanks Mark, I've also got it to work on my iPhone by disabling the iCloud SMTP server and adding p01.
For others going down this road, there's a typo - The correct server name for the workaround server is:
p01-smtp.mail.me.com (not stmp)
I've also got this to work on my OS X accounts by:
Mail -> Preferences -> Accounts
Select the problem account and "Edit SMTP Server List" on the "Outgoing Mail Server" dropdown menu
Add a new server with the + button
Give it a Description (e.g. iCloud Temporary)
Server Name: p01-smtp.mail.me.com
TLS Certificate: None
Click the Advanced Tab
Check Use default ports (25, 465, 587)
Check Use Secure Sockets Layer (SSL)
Authentication: Password
Username: Enter your username
Password: Enter your password
Click OK
Make sure the "faulty account" has the new SMTP server selected. Mine actually says "iCloud Temporary (Offline)" but it appears to work.
Hope this helps someone else...
Also hope that Apple sort out the p04 issue soon!!!