From the description, it is not the issue with HTML in e-mails. That issue will not block e-mail monitoring in any way.
We have not seen this issue reported lately but back in ServiceDesk 7.0 there was a problem in some cases where one e-mail message failing to process could block the e-mail monitoring completely.
What ServiceDesk version is this happening in?
The e-mail inbox used is POP or IMAP?
To find out why this is happening, we need to find out what exactly is there in these "broken" e-mails that actually fails.