Display term description on hover using get_the_term_list

If you mean to add the description of the term as a title attribute which will show on hover then you can “fork” get_the_term_list function to a version that would add the title attribute, something like this should work:

function get_terms_with_desc( $id = 0 , $taxonomy = 'post_tag', $before="", $sep = '', $after="" ){
    $terms = get_the_terms( $id, $taxonomy );
    if ( is_wp_error( $terms ) )
            return $terms;
    if ( empty( $terms ) )
            return false;
    foreach ( $terms as $term ) {
            $link = get_term_link( $term, $taxonomy );
            if ( is_wp_error( $link ) )
                    return $link;
            $term_links[] = '<a href="' . esc_url( $link ) . '" rel="tag" title="'.esc_attr( $term->description ).'">' . $term->name . '</a>';
    }
    $term_links = apply_filters( "term_links-$taxonomy", $term_links );
    return $before . join( $sep, $term_links ) . $after;
}

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