The issue is that the server appears to remove the drafts flag on those messages after a short time, so they are no longer editable. There is something you can do though.
Go to Menu > Accounts and click on the Diagnostics tab for this account.
In the Advanced Options Parameters box, paste this text:
--imap-messages-in-drafts-folder-are-drafts
Then click on Save & Close. This will only work with eM Client version 9.
