A study has found that more than a third of New York State's recipients of Medicaid and other public health insurance programs fail to re-enroll on time, losing coverage even though they remain eligible because of daunting paperwork and other obstacles. The study by the nonprofit New York State Health Foundation said many people were deterred by Medicaid's annual recertification process and that the resulting churning, in which recipients fall off the rolls and then reapply from scratch, costs the state money because it is more inefficient.