After all, since the NYT has a very limited corpus of information, and supposedly people are generating infringing content using their APIs, said hashes can be used to compare whether such content has been generated.
I'd rather have them store nothing, but given the overly broad court order I think this may be the best middle ground. Of course, I haven't read the lawsuit documents and don't know if NYT is requesting far more, or alleging some indirect form of infringement which would invalidate my proposal.
[1] https://ssdeep-project.github.io/ssdeep/index.html
[2] https://joshleeb.com/posts/content-defined-chunking.html