Update: I’ve just found that the problem doesn’t occur when I use the search widget. The navigation search still needs a double-click. Could you please take a look at the CSS settings of the nav search and the result box? Maybe there’s a conflict between the two properties.
That’s strange, it almost looks like a bug with the focus. Any chance you can report it to the plugin author? They might know why this could happen/what we can do to fix it. Maybe just removing focus once the results are shown would fix it.