Exclude duplicated $sticky post from combined loop content

You simply need the is_sticky() conditional check and then act on that. You would probably still need the is_home() and !is_paged() conditional check to only target stikies on the first page of the home page

if ( have_posts() ) :
    while ( have_posts() ) :
        the_post();

        if (    is_sticky()
             && is_home()
             && !is_paged()
        ) {
            // Do what you need to do for stickies
        } elseif ( !is_sticky() ) {
            // Do something else for non-stickies  
        }
    endwhile;
endif;