Spam from NY Times & WSJ top stories notifications on iPhone
The below instructions apply to iOS 10. There will be similar settings on previous versions of iOS, though the steps may differ somewhat.


Open the News app and tap on the Favorites tab at the bottom.
Tap on the bell icon in the top-left corner.
Toggle off any channels that you don't want to receive notifications from. You can also toggle on any channels from which you do want to receive them.
Tap Done in the top-right corner to save.


If the channels (i.e. NYT and WSJ) you're getting notifications from aren't there, or you want to get notifications from channels that aren't present, you may have to temporarily add them as favorites to cause them to appear.


Tap on the Search tab at the bottom. Search for "New York Times" or "Wall Street Journal".
Tap on the channel's name.
Tap the + icon in the top-right corner.
Go back to the Favorites tab and follow the above instructions to turn notifications on or off, as applicable.
Once you tap Done on the notifications screen, you can remove the channels you just added as favorites if you want to do so. To do this, tap Edit in the top-right corner, and tap the X in the upper-left corner of each channel you want to remove.


The type of notifications you can get from a channel appears to be all or nothing for a channel, so unless the channel is split into specific sections, you can't opt just for Business stories, for example. There are also generalized "News Editors' Picks" and "News Top Stories" options. From my (admittedly limited) testing, you can't get notifications for topics (such as Business) that combine many channels into one categorized feed.

If the notifications aren't turned on in the News app, I have no idea where they're coming from. In that case, the next time it happens, take a screenshot (Home + Sleep) and post it in your question to help determine the source.