Display only one level subcategory in wordpress

I’m going to guess here. I assume you are using the Taxonomy Images plugin?

This:

if($term->term_id == $categories_item->term_id) {

Can be changed to:

if($term->term_id == $categories_item->term_id && $term->parent == $categories_item->cat_ID) {

Hereby, you check if the parent ID of the term is the same as the ID of the categories_item.