If it is talking about something called “precinct reporting” (“precinct” being an attributive noun, modifying the noun “reporting”), then “precinct” would be correct.
If it is talking about several precincts performing the action of reporting then it should be “precincts”.
If it is talking about a part of a single precinct performing the action of reporting then it should be “precinct”.
We’ve got [31% of the precincts] reporting in the state of New York.
This would mean that 31% of the precincts have reported and 69% of them have not.
We’ve got 31% of the [precinct reporting] in the state of New York.
Here “precinct reporting” is a compound noun, and the sentence means that 31% of the reporting from all the precincts has arrived. They’re not counting how much they’ve received from just one precinct, but how much they’ve received from the precincts overall.