Hi,
It seems like the plugin shortcode is placed within a menu, that is set to invisible on mobile devices – and since that is not visible, the search is not visible either. I recommend adding the shortcode to the second menu as well, or maybe using a widget or a builder element that is visible only on mobile devices.
Page builders usually have options to change which block is visible on which devices. Perhaps the easiest way is to add a shortcode/HTML block with the plugin shortcode, wherever you need it, that is only visible on mobile devices.