Having the city pre-loaded into the search bar is not a default option within the theme so it would require some customization on your part if you wanted to do this, but it can technically be done as there is no encryption in the theme that would prevent you from making such changes.
The search is going to use the keyword you have typed in for the search, and if you have also specified a location it will look for listings that match the keyword in that location you specify. Unless other locations are within the radius of the original location, then they won't appear in the results. As you can see from the
Vantage demo site, you can adjust the radius when searching. Thanks.