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Assuming this is the default main query, see the example for pre_get_posts, which shows how to exclude a category from the main query:

function wpa78465_exclude_category( $query ) {
    if ( $query->is_home() && $query->is_main_query() ) {
        $query->set( 'cat', '-1,-1347' ); // IDs of categories to exclude
    }
}
add_action( 'pre_get_posts', 'wpa78465_exclude_category' );