Inject post (from specific category) between posts in Loop

I would like to add a post (from specific category) between posts in the WordPress Loop (such as ‘Sponsors’). Example:

P P P S
P P S P
P S P P

How can I achieve this? I am a beginner with coding, so I don’t know enough to modify a Loop on my own. Any Loop Coding Ninjas out there that can provide a solution?

Note that below is my current Loop. It is used to sort posts by price, or by random order:

index.php

<?php while (have_posts() ) : the_post(); ?>
<h2><a href="https://wordpress.stackexchange.com/questions/158133/<?php the_permalink() ?>"><?php the_title(); ?></a></h2>
<?php  the_excerpt(); the_meta ();
endwhile; 

previous_posts_link();
next_posts_link();
?>

functions.php

function my_custom_query($query){
 if ( $query->is_home() && $query->is_main_query() ) {

  $sort= $_GET['sort'];

  if($sort == "A"){
   $query->set( 'orderby', 'rand' );
   $query->set( 'posts_per_page', '2' );
  }

  if($sort == "B"){
   $query->set( 'meta_key', 'price' );
   $query->set( 'orderby', 'meta_value_num' );
   $query->set( 'order', 'DESC' );
   $query->set( 'posts_per_page', '2' );
  }
}
}

add_action( 'pre_get_posts', 'my_custom_query' );

Edit: Update

Birgire’s plugin works! Initially, I had problems getting the plugin to work on my theme. The problem was this piece of code that I use within the Loop in index.php (I use it to call for the custom field to appear).

<?php
    global $wp_query;
    $postid = $wp_query->post->ID;
    echo get_post_meta($postid, 'price', true);
    wp_reset_query();
?>

The Automatic Sponsor Posts Injector:

Here’s one idea based on my answer for the question:
How to show Y number of custom posts after every X normal posts?

I hopefully made it a little bit more useful here on Github, but it may be refined much more (future work).

The SponsorPostsInjector class will help you to automatically inject the sponsor posts into the theme using the filters the_post, loop_start and loop_end.

Activate the plugin and add the following example to your functions.php file to start the injections:

/**
 * Inject a sponsor post after the first post on the home page,
 * and then again for every third post within the main query.
 */

add_action( 'wp', 'my_sponsor_injections' );

function my_sponsor_injections()
{
    if( ! class_exists( 'SponsorPostsInjector' ) ) return;

    // We want the sponsor posts injections only on the home page:
    if( ! is_home()  ) return;

    // Setup the injection:
    $injector = new SponsorPostsInjector( 
        array(
            'items_before_each_inject' => 3,
            'items_per_inject'         => 1,
            'template_part'            => 'content-sponsor',
        ) 
    );

    // Setup the injection query:
    $injector->query(
        array(
            'post_type'  => 'sponsor',
            'tax_query'  => array( 
                array(
                   'taxonomy' => 'country',
                    'terms'   => 'sweden',    
                    'field'   => 'slug', 
                )
            )
        )
    );

    // Inject:
    $injector->inject();
}

where we have created the content-sponsor.php template file in our current theme directory, to control the layout of the injected sponsor posts.

The idea is that this should also take care of the pagination.

You can hopefully adjust this to your needs.