It's been many years ... so this is from 'best recollection' ...
See the '1' in the URL to the image?
That file should be a directory in /moodledata/1/ ... the directory for all files related
to your 1.9 front page. Verify that file is there.
Can you turn on editing on the front page and see if the image is in the site
summary? If so, download the image first, then remove it and save. The re-edit the summary and upload. Now it will reference the image using https.
For each image that causes the mixed content issue, one needs to do the same as described above.
If that's too many to do one at a time, the only other way is to take a mysqldump of
the database. The sql file is asciii so one can edit it via any ASCII editor .... *NOT
MS Word other other word processor*.
Search for http://yoursite/ replace with https://yoursite/ in the entire sql file. Note again that's changing http to https. And you want to use the full URL for your site with the ending '/' to avoid globally changing other URL's that might be in the SQL file.
Then in the mysql client, create a new db for the moodle (call it something like moodle2), use the edited sql file to import into that new moodle2 db.
Last step edit config.php to point to moodle2 DB.
Before doing all that ...
Version 1.9 didn't have a purge cache. But, *if* site is using a PHP accelerator ... which was recommended back then ... you might be able to run an opache php script to refresh what caching it's doing. From command line, php -m might show if an accelerator extension is installed.
https://www.stevejenkins.com/blog/2014/07/my-favorite-zend-opcache-status-scripts/
The OCP – OPcache Control Panel in above does have a 'Reset'.
That's a separate script ... not a moodle plugin/extension/mod etc.
Just upload to the root of the moodle code and access directly with browser.
Once you are done converting, move the script out of web root ... save it somewhere else on server ... like in /root ... that's because anyone discovering the script is there can use it.
Needless to say, 1.9 is very old and it would be wise to think about how to upgrade that site.
'spirit of sharing', Ken