The city pays for some library expenses, but we never see the money. Should we report it?

Yes.  Basically, significant expenses made on the library's behalf by a governmental entity should be reported.  If the city didn't pay for the building directly, the library would have to pay.  Aside from advocacy and shameless money-grubbing, the annual statistical report answers the question "What does it take to run this library for a year."  To not report a significant expense made on the library's behalf is to skew the cost per service to unrealistic numbers.  Have mercy for everyone else who might try to match that cost per service!