The thing is, you didn’t mention that you were using the hamburger from the start.
I can see the item but it is set to display none by css code.

to see a first result bring this to quick css and look:
( a little bit more css to see selectors if influnce )

#top .menu-item-cart.menu-item-avia-special  {
  display: block !important;
}
#top .menu-item-cart.menu-item-avia-special a {
  font-size: 24px !important;
}
.av-burger-overlay-active #top .menu-item-cart.menu-item-avia-special a {
	color: #FFFFFF !important;
}
#top .menu-item-cart.menu-item-avia-special a:hover {
	color: #9d1a18 !important;
}

(click to enlarge:)

_____________________

and don’t forget to replace the “#” by the link of that cart icon

get rid of the class: menu-item in that snippet:

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 =  $items . $cartLink;
  }
  return $items;
}
add_filter( 'wp_nav_menu_items', 'add_cart_icon_to_main_menu', 9999, 2 );

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.

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/

first : if you have a shop – and that is the cart icon for. i guess there are options to show that cart icon in main menu.
_______

One method to do it:

i prefer the hook on next answer !

do you know how to give a custom class to a menu item?

place inside the menu an custom link ( preferably as the last link in your main menu. ), replace the label of that menu-item with your icon. give the custom-class : menu-item-avia-special to it. thats it
Menu-items with that class will be visible – if the burger is active

1) custom link – the later “navigation label” is what you insert to “Link Text”
– in your case it is enough to insert: <span class='av-icon-char' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span>
2) give to that menu-item the custom class: menu-item-avia-special

if you do not see on editing the menu-items a custom class input field. :
on top right Window of your Menus Option Dialog there is a slide out dialog : “Screen Options” – open it and mark all you needed for that:

to change position with f.e. the search icon that is easy … just ask
( in that case it might be useful to have another custom class on that menu-item f.e. in your case menu-item-cart)