Display the 2nd category name of a custom post type without error if its null?

Inside of your loop you can get the terms via wp_get_post_terms function. Then, check if two or more terms exist. If they do, then use the function array_slice to extract the second term that appears in the array:

$term_list = wp_get_post_terms( $post->ID, 'YOUR_TAXONOMY', array('orderby' => 'name', 'order' => 'ASC', 'fields' => 'all') );

$display_term = (count($term_list) > 1) ? array_slice($array, 1, 1) : $term_list;

However, this does not determine how the terms will output from the database. You may want to define the orderby parameter to be something specific so you can predict which term will appear second.

Sources:

• https://codex.wordpress.org/Function_Reference/wp_get_post_terms
• http://php.net/manual/en/function.array-slice.php