Hi,
I’m afraid this is not a plugin related issue. The search icon in the menu, and the container where the plugin is present is not controlled by the plugin, but the menu (I guess as part of the theme). When hovering over the search icon the container appears, but when looses mouse hover focus it closes – that is causing the issue.
I suggest checking if the menu/theme has an option to display the container only if the icon is clicked, or to never close it once it has been opened. Unfortunately I cannot suggest any custom CSS to resolve this, custom CSS cannot fix this, as it is a 3rd party script related issue.