ALBANY — In January 2016, as Gov. Andrew M. Cuomo was preparing his annual State of the State address, some high-placed state officials knew they had a problem. A pair of business executives — partners in multimillion-dollar, private-public partnership deals with the state — were on their way to Albany, anticipating that Mr. Cuomo was going to announce their projects in his speech. But Mr. Cuomo had decided against doing so, worrying the executives. “We need to assure these guys their deals are solid,” Christopher Walsh, an associate vice president at the State University of New York Polytechnic Institute, wrote...