How to rename custom taxonomy URL with extra words

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.