To delete a wish, open it on your wishlist and choose to delete it. Deleting a single wish is permanent, so only delete a wish you no longer want on the list.
Steps
Open the wishlist that has the wish.
Select the wish you want to remove.
Choose Delete and confirm.
Before you delete
Deleting a wish is permanent. There's no undo for a single deleted wish, so double-check before you confirm. And if a wish was reserved, deleting it cancels the reservation — the person who reserved it isn't notified, the reservation is simply cancelled, and the item is no longer on the list. (This is different from removing just a reservation, where the person who reserved is notified.)
You can delete a wish if you have Contributor access or higher on the wishlist. As a Contributor, you can delete wishes you added. See "What's the difference between Full access and Contributor?" in the Shared Wishlists collection for what each role can do.
Want to keep the item but show it's handled? Instead of deleting, the owner or a collaborator can mark the wish as fulfilled. This keeps it on the list but flags it as taken care of — handy when you've received a gift but don't want to remove the record.