Not NY Times here, but another news site (Dutch): www.nos.nl
Almost the same problem, the clipper grabs the cookie notification.
Your Android app has a function 'clip again', which can be invoked from the menu inside each individual note. When using this function the article gets (re)clipped as expected.
This function is not available in the web version, nor in the iOS or OSX applications. So it can only be done from Android ?
Does it use another engine to grab the contents?
Best regards,
Maik