We haven't tested on Windows 8 phone yet, but we do test on iPad, iPhone, and Android (native browsers only).
We're working on making the Clerk Resources application more mobile-friendly in general (responsive design, etc.), and hope to have that in production in the near future.
I can only speak for the Clerk Resources application (the Leader application is a different team, but has a similar menu). Is the URL "lds.org/leader/<something>" or is it "lds.org/mls/<something>" when you experience the problem? Can you post or PM the URL for the screen where you see this problem?
...tapping the item causes the menu to briefly appear. If I do it again, it stays up to allow me to select something...
I haven't experienced the menu briefly appearing, requiring you to click again. Do you mind sharing the name/type of device you're using when you see that behavior?