Assuming you already created the custom field (with ACF Pro) and have added the field to your post.
On your parent X theme navigate to \x\framework\views\integrity and copy the file content-portfolio.php
Then navigate to your child theme's \x-child\framework\views\integrity directory (create it if it does not exist) and paste the copied file.
Open and edit that pasted file and look for this line
<?php x_portfolio_item_project_link(); ?>
That is the project link button, you need to output your ACF button just below that.
To output your ACF button, you need this line:
<p><?php the_field('FIELD_NAME'); ?></p>
Replace the FIELD_NAME with your actual field name.
Save the edited file and that should do it.
ACF | Displaying custom field.
Hope it helps,