Strange result when running Url of cron task in CBSubs Mass expiry settings...

I figured out that if we are logged into the front end as a registered user, it is their ACL/Permissions that determine whether the cron task will run properly. If we make sure that we are not logged into the frontend as a standard registered user, all works well...

That likely means you've something on frontend that only displays to logged in users that's messing with the output buffer. That URL should result in a completely blank page. If any HTML is rendered on it then something is conflicting with it. Could be a Joomla module or Joomla system plugin. You however shouldn't need to run the URLs manually. They're meant to be ran from a CRON Task, which will always be a public user.

You however shouldn't need to run the URLs manually. They're meant to be ran from a CRON Task, which will always be a public user.

I understand that it is meant to be used as a cron job. But sometimes we make changes to content of notification emails and we run the task manually in order to generate messages to check that what we have done is showing up properly. 

Best I can suggest then is open the URL in an incognito browser window. I don't know what more to recommend as the issue is with something on your site impacting the Joomla component output. You would need to find what's doing that.