Hi Jim,
I have worked on a few of these cuckoo clocks and I do remember that one of them had the same issue your having. The count hook that drops down has a tension wire that sits on top of that count hook, It’s about 1 1/2″ long it stems from the opposite plate and It prevents the count hook from prematurely jumping and causing it to advance and strike again mainly on the half hour strike. Thats what it was doing to the one I was working on. Check that to be sure it has the correct tension. Even though it could be another issue try that first.
Take care,
Jim