When you send a gift, the gift will be wrapped (and invisible to other users) until the receiver unwraps the gift.
While the gift is wrapped, you can't delete the gift unless the user hasn't unwrapped it after 1 month since you sent it.
If the other user has already unwrapped the gift, you can set it permanently to "Private", so the gift will not be visible to anyone else, other than you and the receiver. Of course, if the gift was already "Private" when you sent it, you don't need to set it as "Private" when the user unwraps the gift.
To return a wrapped gift or mark an unwrapped gift as "Private", you can go to your Profile -> Gifts -> Sent and find the gift you want to action.