Custom Taxonomy Is Being Pulled into a Page, But It Doesn’t Have A Hyperlink

@Chewy you can try below code snippet.

<p class="cat">
    <?php
    $terms = get_the_terms($post->ID, 'news_categories');
    foreach ($terms as $term) {
        $term_id    = $term->term_id;
        $term_name  = $term->name;
        $term_link  = get_term_link( $term_id );
        echo "<a href="".$term_link."">".$term_name."</a>";
    }
    ?> 
</p>

please chcek and let me know if this work or not.

if not then please ping me I will definitely help you.

Thanks.