When I register my custom post it adds a menu option to the WP Admin Dashboard that users can click and it lists all custom posts of that type. (Please see my image below if I am not making sense about the page I am describing).

That page listing all the custom posts of that type is good and working, but I only want that page to show certain custom posts. I only want to show custom posts whose post_meta my_flag equals true. Everyone of my custom posts has a my_flag post_meta set to true or false. I want my list page to only show custom posts whose my_flag = true.

Is that possible to do when I register the custom post type? Even if it isn’t, how would I achieve this?

$labels = array(
    'name'                => _x( '_foo', 'Post Type General Name', '_foo' ),
    'singular_name'       => _x( 'Foo', 'Post Type Singular Name', '_foo' ),
    'menu_name'           => __( '_foo', '_foo' ),
    'parent_item_colon'   => __( 'Parent _foo', '_foo' ),
    'all_items'           => __( 'Foo Posts', '_foo' ),
    'view_item'           => __( 'View Foo', '_foo' ),
    'add_new_item'        => __( 'Add New Foo', '_foo' ),
    'add_new'             => __( 'Add New', '_foo' ),
    'edit_item'           => __( 'Edit Foo', '_foo' ),
    'update_item'         => __( 'Update Foo', '_foo' ),
    'search_items'        => __( 'Search Foo', '_foo' ),
    'not_found'           => __( 'Not Found', '_foo' ),
    'not_found_in_trash'  => __( 'Not found in Trash', '_foo' ),
);

$args = array(
    'label'               => __( 'foo', '_foo' ),
    'description'         => __( 'My custom posts', '_foo' ),
    'labels'              => $labels,
    'supports'            => array( 'title', 'revisions', 
    'taxonomies'          => array( 'foo' ), 
    'hierarchical'        => false,
    'public'              => true,
    'show_ui'             => true,
    'show_in_menu'        => 'my-plugin-menu',
    'show_in_nav_menus'   => true,
    'show_in_admin_bar'   => true,
    'show_in_rest'        => true,
    'query_var'           => true,
    'menu_position'       => 5,
    'can_export'          => true,
    'has_archive'         => true,
    'exclude_from_search' => false,
    'publicly_queryable'  => true,
    'capability_type'     => 'page'
);

register_post_type('foo', $args);

I set the post_meta like so:

add_post_meta($post_id, 'my_flag', true, true);