You’d be better off adding or modifying the rewrite
variable that’s being passed to your register_taxonomy()
call for the custom taxonomy. You can modify the rewrites to suit your purpose.
For example:
$taxonomy_args = array(
// All the variables and their values...
'rewrite' => array(
'slug' => 'services/home-care',
),
// ...more variables and values...
);
register_taxonomy( 'tax_city', $object_types, $taxonomy_args );
Be sure to flush your site’s permalinks after you make the change.