Custom post type not using it's own single-postname.php

Have you flushed your permalinks?

If not, visit your permalinks page in your dashboard, then, refresh the page you are trying to view.

Let us know if that helps.

Edit:

Upon re-reading your question, actually, no – just because you create a single-post_type.php file does not mean WordPress will use that file with regards to your front-page.php template file.

Your front-page.php file is what is rendering your post. This is your issue.

If you want your single-post_type.php file to render the display of your post type within front-page.php then you can do the following;

<?php
$portfolio_query = new WP_Query(array(
'post_type' => 'headerhome',
'showposts' => 1
));
?>

<?php while ($portfolio_query->have_posts()) : $portfolio_query->the_post(); ?>

<?php get_template_part('single-post_type.php'); ?>
   
<?php endwhile; wp_reset_postdata();?>

But obviously make sure that single-post_type.php does not contain a loop/query within itself for if it does then you should remove the outer while/endwhile statements contained within front-page.php

Edit:

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