Icon Next to Burger / All previous topics won’t work

In fact, you could do it with a hook as well. – but i would use a wordpress hook on that.
The main navigation menu theme_location is avia.
( you see that you can use that hook also to other menu locations :
avia(main-menu), avia2( top-menu), avia3(footer-menu) )

function add_cart_icon_to_main_menu( $items, $args ){
  if ($args->theme_location == 'avia'){
    $cartLink = '<li class="menu-item menu-item-cart menu-item-avia-special" role="menuitem"><a aria-label="Cart" href="#" rel="nofollow" aria-hidden="false" data-av_icon="" data-av_iconfont="entypo-fontello" alt="Cart" ><span class="avia_hidden_link_text avia-menu-text">Cart</span></a></li>';
    $items =  $items . $cartLink;
  }
  return $items;
}
add_filter( 'wp_nav_menu_items', 'add_cart_icon_to_main_menu', 9999, 2 );

replace that “#” in the snippet with your prefered link

this will insert a new menu-item to the main nav. But: it will be the very last menu-item even behind the search icon and even behind the hamburger icon itself.
Same here moving it to the right position is easy with that extra class from snippet ( menu-item-cart)

function move_cart_icon_before_burger() { 
?>
<script>
(function($){
  $('li.menu-item-cart').insertBefore($('li.av-burger-menu-main'));
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'move_cart_icon_before_burger');

to see that solution used : https://webers-testseite.de/

Please remove the other solution trials – do not use a combined solution – and only use that: https://kriesi.at/support/topic/icon-next-to-burger-all-previous-topics-wont-work/#post-1424623
And copy paste the code from there to your child-theme functions.php.
After doing that we can discuss when or when not the icon is visible.

By the way – if you are shure that you only need the burger menu from the beginning, you can change that snippet to that – and forget the moving menu item ( snippet 2 )
( because this line $items = $cartLink . $items will do the job. – first insertion then the other existing links )

function add_cart_icon_to_main_menu( $items, $args ){
  if ($args->theme_location == 'avia'){
    $cartLink = '<li class="menu-item-cart menu-item-avia-special" role="menuitem"><a aria-label="Cart" href="#" rel="nofollow" aria-hidden="false" data-av_icon="" data-av_iconfont="entypo-fontello" alt="Cart" ><span class="avia_hidden_link_text avia-menu-text">Cart</span></a></li>';
    $items =  $cartLink . $items ;
  }
  return $items;
}
add_filter( 'wp_nav_menu_items', 'add_cart_icon_to_main_menu', 9999, 2 );

I will never understand why the page links are not published here. Because assistance would be a lot easier (for us participants) if we could see the page in question.