Verify if a category is the child of another category

Yes, there are two functions you can use:

For the default category taxonomy, you can use cat_is_ancestor_of():

// Retrieve category object by slug:
$parent_cat = get_category_by_slug( 'clothes' );  // term object
$child_cat  = get_category_by_slug( 't-shirts' ); // term object

/* Or retrieve category ID by name:
$parent_cat = get_cat_ID( 'Clothes' );  // term ID
$child_cat  = get_cat_ID( 'T-Shirts' ); // term ID
*/

if ( cat_is_ancestor_of( $parent_cat, $child_cat ) ) {
    echo 't-shirts is child of clothes';
}

For custom taxonomies, you would use term_is_ancestor_of():

// Set the custom taxonomy's slug:
$taxonomy = 'my_taxonomy';

// Retrieve term object by slug:
$parent_cat = get_term_by( 'slug', 'clothes', $taxonomy );
$child_cat  = get_term_by( 'slug', 't-shirts', $taxonomy );
/* Or name, if you want:
$parent_cat = get_term_by( 'name', 'Clothes', $taxonomy );
$child_cat  = get_term_by( 'name', 'T-Shirts', $taxonomy );
*/

if ( term_is_ancestor_of( $parent_cat, $child_cat, $taxonomy ) ) {
    echo 't-shirts is child of clothes';
}

And note that cat_is_ancestor_of() is really a wrapper around (i.e. it uses) term_is_ancestor_of() which accepts either term ID or object, so for examples, you could do cat_is_ancestor_of( 1, 2 ) and term_is_ancestor_of( 3, 4, 'my_taxonomy' ) whereby the numbers like 1 and 3 are term IDs.