See, you need to update you $item_output variable. fontawesome outputs the icons in this format <i class="fa fa-angle-right"></i>.

you need to add this <i> within your <a> like this.

$item_output = "$args->before<a $attributes>$args->link_before<i class='$variable_which_stores_fontawesomme_value'></i>$title</a>" . "$args->link_after$args->after";

$variable_which_stores_fontawesomme_value should be equal to fa fa-angle-right.

See, you need to update you $item_output variable. fontawesome outputs the icons in this format <i class="fa fa-angle-right"></i>.

you need to add this <i> within your <a> like this.

$item_output = "$args->before"
    . "<a $attributes>"
        . "$args->link_before"
        . "<i class='$variable_which_stores_fontawesomme_value'></i>"
        . "$title"
    . "</a>"
    . "$args->link_after"
    . "$args->after";

$variable_which_stores_fontawesomme_value should be equal to fa fa-angle-right.