The carriers TNT - are responsible for any damage caused to the goods after they collected them from your premises and whilst in their custody, right up to and including handling at the point of delivery at the other end. (Did the recipient sign as damaged ?? ensure you also keep that email the recipient has sent... informing you the goods were damaged)
You need the recipient to record the extent of the damage caused by the carrier and if possible photograph it. Meantime contact the carriers requesting a damaged goods claim form, then provide the carrier with all details of the damage and the amount of loss they have caused.