After updating the order in the shopping cart the order for Klarna need to be updated, this is done automatically in the IPaymentWidget.GetWidget method for the Klarna payment widget (implemented in KlarnaPaymentWidgetController).
If you not refreshing the widget you anyway need to update the order information that is sent and stored for Klarna, see the above method with the call of method klarnaCheckout.CreateOrUpdateKcoOrder that you manually need to invoke.
Note: File and method names is from Acceleratorn 6.3 but I think it should be the same in your version also.