acosmin

Hey, sorry, but this is starting to feel like custom work. I tried helping but I can only give support for bugs regarding the default functionality of the theme.

You can leave this topic opened, maybe someone will help you.

You can also try:
https://wordpress.org/support/forum/themes-and-templates
http://jobs.wordpress.net/

Change your index contents with this:

<?php
/* ------------------------------------------------------------------------- *
 *	Index template
/* ------------------------------------------------------------------------- */
?>
<?php get_header(); ?>

<section class="container<?php ac_mini_disabled() ?> clearfix">

    <?php get_sidebar( 'browse' ); ?>

    <div class="wrap-template-1 clearfix">

    <section class="content-wrap" role="main">

        <?php
			if ( of_get_option( 'ac_show_slider' ) && is_front_page() && !is_paged() && ac_featured_posts_count() > 2 ) {
				get_template_part( 'featured-content' );
			}
		?>

    	<div class="posts-wrap clearfix">

        <?php
                        $args = array( 'cat' => -2691, 'posts_per_page' => 6 );
                        $new_query = new WP_Query( $args );
			if ( $new_query->have_posts() ) :
				while ( $new_query->have_posts() ) :
					$new_query->the_post();
					get_template_part( 'post-templates/content' );
				endwhile;
			else :
				get_template_part( 'post-templates/content', 'no-articles' );
			endif;
		?>

        </div><!-- END .posts-wrap -->

        <?php ac_paginate(); wp_reset_postdata(); ?>

    </section><!-- END .content-wrap -->

    <?php get_sidebar(); ?>

    </div><!-- END .wrap-template-1 -->

</section><!-- END .container -->

<?php get_footer(); ?>

Replace all index.php contents with this:

<?php
/* ------------------------------------------------------------------------- *
 *	Index template
/* ------------------------------------------------------------------------- */
?>
<?php get_header(); ?>

<section class="container<?php ac_mini_disabled() ?> clearfix">

    <?php get_sidebar( 'browse' ); ?>

    <div class="wrap-template-1 clearfix">

    <section class="content-wrap" role="main">

        <?php
			if ( of_get_option( 'ac_show_slider' ) && is_front_page() && !is_paged() && ac_featured_posts_count() > 2 ) {
				get_template_part( 'featured-content' );
			}
		?>

    	<div class="posts-wrap clearfix">

        <?php
                        $paged = get_query_var( 'paged' ) ? get_query_var( 'paged' ) : 1;
                        $args = array( 'cat' => -2691, 'paged' => $paged );
                        $new_query = new WP_Query( $args );
			if ( $new_query->have_posts() ) :
				while ( $new_query->have_posts() ) :
					$new_query->the_post();
					get_template_part( 'post-templates/content' );
				endwhile;
			else :
				get_template_part( 'post-templates/content', 'no-articles' );
			endif;
		?>

        </div><!-- END .posts-wrap -->

        <?php ac_paginate(); wp_reset_postdata(); ?>

    </section><!-- END .content-wrap -->

    <?php get_sidebar(); ?>

    </div><!-- END .wrap-template-1 -->

</section><!-- END .container -->

<?php get_footer(); ?>

Try:

<?php
                        $paged = get_query_var( 'paged' ) ? get_query_var( 'paged' ) : 1;
                        $args = array( 'cat' => 3, 'paged' => $paged );
                        $new_query = new WP_Query( $args );
			if ( $new_query->have_posts() ) :
				while ( $new_query->have_posts() ) :
					$new_query->the_post();
					get_template_part( 'post-templates/content' );
				endwhile;
			else :
				get_template_part( 'post-templates/content', 'no-articles' );
			endif;
wp_reset_postdata();
		?>

You want to exclude posts that are from that category or you don’t want to show that category name bellow posts?

Can you make a printscreen and show me what you want to hide?

Instead of:
$new_query = new WP_Query( 'cat=-2691' );
try:
$new_query = new WP_Query( 'category__not_in' => 2691 );

Do the posts on page 4 have more than one category?

can you post a link to your website?