Ok, I got it a little further.
If I add my new database column to the getProducts function in the Cart.php, it works. Now it looks like that:
// Build SELECT
$sql->select('cp.`id_product_attribute`, p.`packing_unit_for_groups`, cp.`id_product`, cp.`quantity` AS cart_quantity, cp.id_shop, pl.`name`, p.`is_virtual`,
pl.`description_short`, pl.`available_now`, pl.`available_later`, product_shop.`id_category_default`, p.`id_supplier`,
p.`id_manufacturer`, product_shop.`on_sale`, product_shop.`ecotax`, product_shop.`additional_shipping_cost`,
product_shop.`available_for_order`, product_shop.`price`, product_shop.`active`, product_shop.`unity`, product_shop.`unit_price_ratio`,
stock.`quantity` AS quantity_available, p.`width`, p.`height`, p.`depth`, stock.`out_of_stock`, p.`weight`,
p.`date_add`, p.`date_upd`, IFNULL(stock.quantity, 0) as quantity, pl.`link_rewrite`, cl.`link_rewrite` AS category,
CONCAT(LPAD(cp.`id_product`, 10, 0), LPAD(IFNULL(cp.`id_product_attribute`, 0), 10, 0), IFNULL(cp.`id_address_delivery`, 0)) AS unique_id, cp.id_address_delivery,
product_shop.advanced_stock_management, ps.product_supplier_reference supplier_reference, IFNULL(sp.`reduction_type`, 0) AS reduction_type');
Can somebody explain to me, how to override that class with as less duplicated code as possible? In the moment I'm overriding the whole function (getProducts).