Getting child category id (help with improving my code)

Starting from your parent category slug you can use the following code to get the child categories:

$parent = get_category_by_slug('winners-2018');

$catz = get_categories(
    array( 'parent' => $parent->cat_ID )
);

foreach ($catz as $cat_archive) {
    // No need for get_category(); because get_categories() already returns category objects
    ...
}