Many hospitals in downtrodden areas of New York City and across the state are faltering, raising concerns that a wave of closings will deprive poor people of badly needed care. A three-month investigation by The New York Times into Wyckoff, based on dozens of interviews and an examination of internal documents, offers a sobering portrait of how one such hospital has been undermined by the very people entrusted to run it. The hospital all but defaulted on its $109 million in state-secured bonds, forcing the taxpayers to cover $10 million due to bondholders before the state agreed in May to defer the hospital's overdue payments.