you can use the pre_get_posts action hook to remove the specific product.
add_action( 'pre_get_posts', 'wp8774_pre_get_posts' );
function wp8774_pre_get_posts( $q ) {
if ( ! $q->is_main_query() ) return;
if ( ! $q->is_post_type_archive() ) return;
if ( ! is_admin() && is_shop() ) {
$q->set( 'tax_query', array(array(
'taxonomy' => 'product_cat',
'field' => 'slug',
'terms' => array( 'private-clients', 'charity'), // Don't display products in the private-clients category on the shop page
'operator' => 'NOT IN'
)));
}
remove_action( 'pre_get_posts', 'wp8774_pre_get_posts' );
}