If a card says: Response- As it enters the discard the pile (or when it dies) draw a card

It is wounded due to loosing Combat struggle. Since the card enters discard pile during the Icon Struggle Phase, does its response still gets to be resolved at then end of the Struggle?(Although the card has gone to the discard phase already)

Yep! The Response will be triggered after the icon struggles.

It still happens, it just has to wait until the story struggles are completed.

There is a special response slot for such cards after all stories have resolved. See timing chart in rulebook.