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/**
 * Update the order meta with field value
 */
add_action( 'woocommerce_checkout_update_order_meta', 'my_custom_checkout_field_update_order_meta' );

function my_custom_checkout_field_update_order_meta( $order_id ) {
    if ( ! empty( $_POST['my_field_name'] ) ) {
        update_post_meta( $order_id, 'My Field', sanitize_text_field( $_POST['my_field_name'] ) );
    }
}