Hi Paul,
I can help point you in the right direction on this customization.
If you follow our themer’s guide:
https://theeventscalendar.com/support/documentation/events-calendar-themers-guide/
You will want to move these two files to your theme to modify:
\the-events-calendar\views\month\single-event.php
\the-events-calendar\views\month\tooltip.php
The Single Event File there has an explanation of how the toolips are made and how you can add fields. It does involve some javascript and php to do.
We can help out on specific questions, but it will be beyond the support to troubleshoot or complete the customization for you on this forum.
Here are some my tickets that might help you get going on this:
Add custom fields to tooltip in month view
Add venue to week view tooltip
Let me know if you have any follow up questions after you get started and I can try to help out.
Thanks