How do I display an article using a WordPress custom field?

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Yes, and you can use the meta_key along with meta_value parameter. Secondly, you’re going to need two queries when there are no featured posts:

  1. Query the featured articles/posts:

    $q = new WP_Query( [
    'post_type'      => 'post',
    'meta_key'       => 'featured',
    'meta_value'     => '1',
    'posts_per_page' => 3,
    'no_found_rows'  => true,
] );

  2. Query random posts:

    $q = new WP_Query( [
    'post_type'      => 'post',
    'orderby'        => 'rand',
    'posts_per_page' => 3,
    'no_found_rows'  => true,
] );

Here’s an example:

$q = new WP_Query( [
    'post_type'      => 'post',
    'meta_key'       => 'featured',
    'meta_value'     => '1',
    'posts_per_page' => 3,
    'no_found_rows'  => true,
] );

// No featured posts.
if ( ! $q->have_posts() ) {
    $q = new WP_Query( [
        'post_type'      => 'post',
        'orderby'        => 'rand',
        'posts_per_page' => 3,
        'no_found_rows'  => true,
    ] );
    echo 'Displaying random posts.'; // test
} else {
    echo 'Displaying featured posts.'; // test
}

if ( $q->have_posts() ) {
    while ( $q->have_posts() ) {
        $q->the_post();

        // Display the post.
        the_title( '<h3>', '</h3>' );
        //...
    }
}

PS: When pagination is not needed, you should always set the no_found_rows to true. Also, I’m assuming you’d use the standard custom fields editor or a plugin like ACF, to add/manage the custom field to/for your posts.