I'd recommend you ask them directly from their JSD portal: https://productsupport.adaptavist.com/servicedesk/customer/portals
From what I can tell, I think you are right that the public API is lacking a PUT option to just add test cases.
But if you feel adventurous (and the fine adaptavist folks may not like what I'm about to tell you), you can examine the calls made by your browsers when you add test cases to an existing test run. You will see that it's possible to PUT to /rest/tests/1.0/testrunitem/bulk/save.
The payload looked like this when I added 2 test to a run that already had 2:
I suspect you can omit the updatedTestRunItemsIndexes if you add your new test cases with new index numbers (rather than at the top like this did).
This non-public API, in my experience, can be used with that same authentication as the public one.
Of course, they are not guaranteed to be consistent from version to version since they are designed to be used internally by the app. So if you have problems with it, then Adaptavist support may not be willing to help you.
Thank you so much Peter ... Will try the JSD portal ...
yes, have tried inspect, found the same one as you have mentioned... when I have tried adding to existing RUN, it was adding only one test case though my pay load was more than one... I couldn't continue on this since it was non-public api and as you too mentioned, there wont be support if they deprecate it later in future...
will try my luck with JSD portal :-) ..
Thanks once again