Hi Eve KJ,
The problem is that the menu has a higher z-index CSS attribute number, so it appears over the results.
The fix is very simple, with this custom CSS code you can force the search results list to a higher z-index as the menu, so it appears over it:
[html]div[id*=ajaxsearchprores] {
z-index: 99999 !important;
}[/html]
Apply this code to your theme custom CSS field (if it supports it) or use the custom CSS field on the search plugin back-end.