On the Blogpost I have categories showing, is there a way to exclude a catgegory like ‘Media’?

Question

Add this simple check to your foreach loop:

<?php
$categories = get_the_category();
foreach($categories as $cat):
    if ( $cat->name == 'Media' ) continue;
?>

    <a href="<?php echo get_category_link($cat->term_id); ?>">
        <?php echo $cat->name; ?>
    </a>

<?php endforeach; ?>

You can check the category name against the list of categories:

<?php
$categories = get_the_category();
foreach($categories as $cat):
    if ( in_array( $cat->name, array( 'Media', 'Other Category 1', 'Other Category 2' ) ) ) continue;
?>

    <a href="<?php echo get_category_link($cat->term_id); ?>">
        <?php echo $cat->name; ?>
    </a>

<?php endforeach; ?>