classified search in dashboard

Right now i am afraid this is not really possible as the WP_List_Table class (https://developer.www.ads-software.com/reference/classes/wp_list_table/) does not have any filter or action in search_box() method, so it is not possible to insert there an additional field.

When i will find some free time i will see if there is some other way to insert additional parameters in the wp-admin search.

I found this:

https://wordpress.stackexchange.com/questions/11758/extending-the-search-context-in-the-admin-list-post-screen

Is it a possible solution?

Yes, you can customize the code available there and apply it to WPAdverts search.