The News "application" that is present on newly created sites is really just a content folder. So you can just create a new one (or delete it and not worry about it).

As with any page/folder in Trash, you recover it using Cut and Paste. i.e., go to the page in Trash and click Cut, then go to where you want to put the item and click Paste.

(the Trash folder doesn't remember where the page used to be before you deleted it, hence you need to manually Cut and Paste it back to wherever you want it)

Thanks

Joe