How can I add content to a post that only shows in single-post mode?

Looking at your specific example … You probably want to use a shortcode. Something like:

add_shortcode('hide_from_summary','wpse_hide_stuff_here');
function wpse_hide_stuff_here($atts,$content){
     global $post;  // gives access to the post, so you can determine mode

     // don't know if you'll want/need $atts, so ... generic stuff here
     $atts = shortcode_atts( array(
        'foo' => 'no foo'
     ), $atts, 'bartag' );

    // default output of this shortcode is blank (hidden)
    $out="";

    //Only show your content if ... you are viewing the post by itself
    if (is_a($post, 'WP_Post') && is_single($post)) $out = $content;

    return $out;
}

You can add this shortcode to your functions.php or a plugin. Then, apply it in the following way.

[hide_from_summary]
  <p>THIS TEXT ONLY SHOWS IN SINGLE POST MODE SOMEHOW</p>
[/hide_from_summary]