Exclude a specific category link from archive / front page

Assuming that you are using get_categories to print the category links, there is an exclude parameter. You need to use IDs, so you will need to convert the slug with get_term_by

$cid = get_term_by('slug','update','category');
// var_dump($cid);
get_categories(
  array(
    'exclude' => $cid->term_id
  )
);