Have different number of posts on first page

EDIT – ANSWER REVISITED

I’ve being working on another solution which is actually better the original answer. This does not involve any custom query and I think for all purposes my original answer can be dropped but kept for informational purposes

I still believe you are on the homepage and will also treat this as such. So this is my new solution

STEP 1

Remove the custom query from the homepage and replace it with the default loop

<?php

        if ( have_posts() ) :
            // Start the Loop.
            while ( have_posts() ) : the_post();

                ///<---YOUR LOOP--->

            endwhile;

                //<---YOUR PAGINATION--->   

            else : 

                //NO POSTS FOUND OR SOMETHING   

            endif; 

    ?>

STEP 2

Use pre_get_posts to alter the main query to add your custom taxonomy to the main query to display on the home page.

STEP 3

Now, get the posts_per_page option set from the back end (which I assume is 2) and also set your offset which we are going to use. That will be 1 as you will need 3 posts on page one and 2 on the rest

$ppg = get_option('posts_per_page');
$offset = 1;

STEP 4

On page one, you’ll need to add the offset to posts_per_page will add up to 3 to get your three posts on page one.

$query->set('posts_per_page', $offset + $ppp);

STEP 5

You must apply your offset to all subsequent pages, otherwise you will get a repetition of the last post of the page on the next page

$offset = $offset + ( ($query->query_vars['paged']-1) * $ppp );
$query->set('posts_per_page',$ppp);
$query->set('offset',$offset); 

STEP 6

Lastly, you need to subtract your offset from found_posts otherwise your pagination on the last page will be wrong and give you a 404 error as the last post will be missing due to the incorrect post count

NOTE: This piece of code broke pagination on the search page. This is now fixed, see the updated code

function homepage_offset_pagination( $found_posts, $query ) {
    $offset = 1;

    if( $query->is_home() && $query->is_main_query() ) {
        $found_posts = $found_posts - $offset;
    }
    return $found_posts;
}
add_filter( 'found_posts', 'homepage_offset_pagination', 10, 2 );

ALL TOGETHER

This is how your complete query will look like that should go into functions.php

function tax_and_offset_homepage( $query ) {
  if ($query->is_home() && $query->is_main_query() && !is_admin()) {
    $query->set( 'post_type', 'my_post_type' );
    $query->set( 'post_status', 'publish' );
    $query->set( 'ignore_sticky_posts', '-1' );
    $tax_query = array(
        array(
            'taxonomy' => 'my_taxo',
            'field' => 'slug',
            'terms' => array('slug1', 'slug2', 'slug3')
        )
    );
    $query->set( 'tax_query', $tax_query );
    $ppp = get_option('posts_per_page');
    $offset = 1;
    if (!$query->is_paged()) {
      $query->set('posts_per_page',$offset + $ppp);
    } else {
      $offset = $offset + ( ($query->query_vars['paged']-1) * $ppp );
      $query->set('posts_per_page',$ppp);
      $query->set('offset',$offset);
    }
  }
}
add_action('pre_get_posts','tax_and_offset_homepage');

function homepage_offset_pagination( $found_posts, $query ) {
    $offset = 1;

    if( $query->is_home() && $query->is_main_query() ) {
        $found_posts = $found_posts - $offset;
    }
    return $found_posts;
}
add_filter( 'found_posts', 'homepage_offset_pagination', 10, 2 );