How to display custom taxonomy term specific post?

Refer Link for Your Knowledge Template Hiearchy for a more detailed break down of how WordPress chooses the template

For a taxonomy term slug (‘sports’ your example) in the taxonomy (e.g. ‘News’) WordPress will try to use the following
templates (in this order)

taxonomy-{taxonomy}-{slug}.php
taxonomy-{taxonomy}.php
taxonomy.php
archive.php
index.php

Modify Your texonomy Code snippet below :

function my_post_type() {    

    $args = array(
        'labels' => array(
            'name' => 'News',
            'singular_name' => 'News',
            'add_new' => 'Add New News',
            'add_new_item' => 'Add New News',
            'edit_item' => 'Edit News',
            'new_item' => 'New News',
            'all_items' => 'All News',
            'view_item' => 'View News',
            'search_items' => 'Search News',
            'not_found' =>  'No News Found',
            'not_found_in_trash' => 'No News found in Trash', 
            'parent_item_colon' => '',
            'menu_name' => 'News',
        ),
        'public' => true,
        'has_archive' => true,
        'menu_icon' => 'dashicons-admin-site-alt3',
        'supports' => array('title','editor','thumbnail'),
        'capability_type' => 'post',

    );   

    register_post_type('news', $args);

}    
add_action('init', 'my_post_type');

function my_taxonomy() {
    $args = array(
        'labels' => array(
            'name' => 'News Category',
            'singular_name' => 'News Category',
        ),
        'public' => true,
        'hierarchical' => true,
    );

    register_taxonomy('news_cat', array('news'),$args);

}

add_action('init', 'my_taxonomy');

For your ‘sports’,’games’,’politics ‘ taxonomy term page, WordPress will use

Put this file in your theme root directory

taxonomy-news_cat-sports.php
taxonomy-news_cat-games.php
taxonomy-news_cat-politics.php

Permalinks
You have also specified an url rewrite, so assuming the rewrite rules have been flushed and there isn’t a clash, the following should also work

www.example.com/news_cat/games

Note : Don’t Forget To update Permalinks Otherwise Categoty term page redirect to 404 page

taxonomy-news_cat-sports.php

<?php 
get_header();

 ?>

<div id="content-sidebar-wrap">

    <div class="wrap">
        <main class="content"> 
            <?php
                $case_cat_slug = get_queried_object()->slug;
                $case_cat_name = get_queried_object()->name;
            ?>
                <h2><?php echo $case_cat_name; ?></h2>
            <?php
                $al_tax_post_args = array(
                    'post_type' => 'news', // Your Post type Name that You Registered
                    'posts_per_page' => 999,
                    'order' => 'ASC',
                    'tax_query' => array(
                        array(
                            'taxonomy' => 'news_cat',
                            'field' => 'slug',
                            'terms' => $case_cat_slug
                        )
                    )
                );
                $al_tax_post_qry = new WP_Query($al_tax_post_args);

                if($al_tax_post_qry->have_posts()) :
                   while($al_tax_post_qry->have_posts()) :
                        $al_tax_post_qry->the_post();
                        echo '<div class="post-excerpt">'; 
            ?>
                        <h2 class="entry-title" itemprop="headline"><a href="<?php the_permalink(); ?>" class="entry-title-link"><?php  the_title(); ?></a></h2>
                        <div class="entry-content"> <?php the_title();  ?> </div>

                        </div>
            <?php
                    endwhile;
                    endif;

            ?>
        </main>         

    </div>  
</div>          

<?php

get_footer();   ?>