After installing the android version on my phone and setting up my hotmail on my phone, the eM client desktop version 10 can successfully send emails.
Great you can now send mail with V10. However adding a hotmail account to a mobile wouldn’t normally fix desktop mail sending problems as its just another way to access the same IMAP account. I suspect it was an SMTP server glitch
However if you get that problem again sending mail via the desktop, try closing eM Client and then installing the latest V10 update (10.1.4588) via the release history page and see if that fixes it. eM Client recently did an update for the desktop for Microsoft sending issues from reading a recent thread.
Now if you still have sending mail problems, then suggest next to try “removing and readding your hotmail account” via “Menu / Accounts” and see if that then fixes the sending mail issues. As might be a possible corrupted mail account.
Note: Before removing make a manual backup via “Menu / Backup” incase you need to restore for any reason. You can see when the backup is complete in Show Operations via clicking the drop-down on the right of Refresh top left.
Lastly if you still have sending mail issues with MS Hotmail, Live or Outlook accts using the latest eM Client V10 and getting sending errors in your log such as " eg: Found invalid data while decoding or “Sync state not found” errors, then see the following eM Client support page where there is a modification you can do to then send mail “via the normal SMTP server” and not use AirSync.
Note: If you do the above eM Client support page modification and have disabled the SMTP service as in my screenshot further up, “re-enable that”.