When you delete notes in Notes Writer the files are moved into the "Trash" location.
Moving items to Notes Writer's Trash doesn’t delete them immediately from your device. You have to empty the Trashcan manually. This fail-safe measure comes in handy when you accidentally delete a note.
If you want to empty the Trash, go into the "Trash" location (Screenshot 1) then tap the "Select" button and select one or more documents. You can permanently delete or restore them if you tap the "Folder" or the "Delete" buttons (Screenshots 2 and 3).
You can also delete all the documents with the "Empty Trash" button available on the bottom toolbar.