Supongo queestofuepesado,peropara responder a supregunta original,recopilétodos los ID de laspublicacionesen unamatrizen elprimer ciclo yexcluíesaspublicaciones del segundo ciclo usando 'post__not_in' queespera una serie de ID depublicación

 <?php
$args1 = array('category_name' => 'test-cat-1', 'order' => 'ASC');
$q1 = new WP_query($args);
if($q1->have_posts()) :
$firstPosts = array();
    while($q1->have_posts()) : $q1->the_post();
        $firstPosts[] = $post->ID; // add post id to array
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
/****************************************************************************/
// array of post id's collected in first loop, can now be used as value for the 'post__not_in' parameter in second loops query $args
$args2 = array('post__not_in' => $firstPosts, 'order' => 'ASC' );
$q2 = new WP_query($args2);
if($q2->have_posts()) :
    while($q2->have_posts()) : $q2->the_post();
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
?>
 

Elprimer ciclomuestratodas laspublicacionesen una categoría y recopila lasidentificaciones de laspublicacionesen unamatriz.

El segundo ciclomuestratodas laspublicaciones,excluyendo las delprimer ciclo.

I suppose this was heavy, but to answer your original question, I've collected all of the posts id's in an array in the first loop, and excluded those posts from the second loop using 'post__not_in' which expects an array of post id's

<?php
$args1 = array('category_name' => 'test-cat-1', 'order' => 'ASC');
$q1 = new WP_query($args);
if($q1->have_posts()) :
$firstPosts = array();
    while($q1->have_posts()) : $q1->the_post();
        $firstPosts[] = $post->ID; // add post id to array
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
/****************************************************************************/
// array of post id's collected in first loop, can now be used as value for the 'post__not_in' parameter in second loops query $args
$args2 = array('post__not_in' => $firstPosts, 'order' => 'ASC' );
$q2 = new WP_query($args2);
if($q2->have_posts()) :
    while($q2->have_posts()) : $q2->the_post();
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
?>

The first loop displays all posts in a category, and collects the post id's into an array.

The second loop displays all posts, excluding posts from the first loop.

Elparámetro queestábuscandoes post__not_in (Kaisertiene unerrortipográficoen su respuesta).Entoncesel códigopodría ser como:

 <?php
$my_query = new WP_Query(array(
    'post__not_in' => array(278),
    'post_type' => 'case-study',
    'paged' => $paged,
));
while ($my_query->have_posts()) : $my_query->the_post(); endwhile;
 

WP_Querypost__not_in documentación

The parameter you are looking for is post__not_in (kaiser has a typo in his answer). So the code could be like:

<?php
$my_query = new WP_Query(array(
    'post__not_in' => array(278),
    'post_type' => 'case-study',
    'paged' => $paged,
));
while ($my_query->have_posts()) : $my_query->the_post(); endwhile;

WP_Query post__not_in documentation

Tienes que definirel argumento post__not_in comomatriz.Inclusopor un solo valor.Y,porfavor,no sobrescriba las variables centralesglobales conelementostemporales.

 <?php
$query = new WP_Query( array(
    'post_type'    => 'case-study',
    'paged'        => $paged,
    'post__not_in' => array( 1, ),
) );
if ( $query->have_posts() ) {
while ( $query->have_posts() ) {
    $query->the_post();

    // do stuff

} // endwhile;
} // endif;
?>
 

You have to define the post__not_in arg as array. Even for a single value. And please don't overwrite global core variables with temporary stuff.

<?php
$query = new WP_Query( array(
    'post_type'    => 'case-study',
    'paged'        => $paged,
    'post__not_in' => array( 1, ),
) );
if ( $query->have_posts() ) {
while ( $query->have_posts() ) {
    $query->the_post();

    // do stuff

} // endwhile;
} // endif;
?>

Códigos alternativos;

Excluirpublicaciones de categoría

 <?php
add_action('pre_get_posts', 'exclude_category_posts');
function exclude_category_posts( $query ) {
    if($query->is_main_query() && $query->is_home()) {
        $query->set('cat', array( -22, -27 ));
    }
}
 

Eliminarpublicaciones de lapágina deinicio

 <?php
add_action('pre_get_posts', 'wpsites_remove_posts_from_home_page');
function wpsites_remove_posts_from_home_page( $query ) {
    if($query->is_main_query() && $query->is_home()) {
        $query->set('category__not_in', array(-1, -11));
    }
}
 

Alternative codes;

Exclude category posts

<?php
add_action('pre_get_posts', 'exclude_category_posts');
function exclude_category_posts( $query ) {
    if($query->is_main_query() && $query->is_home()) {
        $query->set('cat', array( -22, -27 ));
    }
}

Remove posts from homepage page

<?php
add_action('pre_get_posts', 'wpsites_remove_posts_from_home_page');
function wpsites_remove_posts_from_home_page( $query ) {
    if($query->is_main_query() && $query->is_home()) {
        $query->set('category__not_in', array(-1, -11));
    }
}