Display posts of only a certain category in WP Admin section?

You can filter the posts list by appending ?category_name=xx to the admin posts list URL, and you can add a submenu page with that URL as the target via add_submenu_page:

add_action( 'admin_menu', 'wpd_admin_menu_item' );
function wpd_admin_menu_item(){
    add_submenu_page(
        'edit.php',
        'Page title',
        'Menu item title',
        'edit_posts', 
        'edit.php?category_name=somecat'
    );
}