OK - After researching this for a few days, it appears this is a common issue.
However, the standard reply -- "changing the timeout setting in wp-includes/class-http.php from 5 seconds to 30" is not working for me.
I have also deleted all of my plugins one at a time and replaced them to see if that was the issue. It is not.
Also -- this is an intermittent issue. Half of the time, everything works normally. Half of the time, it does not.
This occurs when searching for new plugins or themes and when using askimet or wordpress stats.
Can anyone help with alternative possible solutions?