Sure...
you can log in to Google (it's free if you don't already have a Google account)
There's a variety of search settings that you can look at.
Or, you can choose to let Google's guesses work in your favor. It makes typing much faster, as you'll often get good results as soon as you type 3 or 4 characters, or a couple of words.
Or, you can also choose to ignore that list.
If, instead, you are concerned about YOUR own search history, and it always appears in the Google search list, then that is usually a simple choice of "Clear Recent Searches", which is found by clicking on the icon in an empty search box.
You may also get some change by removing Firefox Caches, in your Caches folder.
You could also check for relevant entries in Firefox's Cookies list.
Perhaps someone else will read this, and have some input on that, too.