Motivated by the pressing need for suicide prevention through improving behavioral healthcare, we use medical claims data to study the risk of subsequent suicide attempts for patients who were hospitalized due to suicide attempts and later discharged. Understanding the risk behaviors of such patients at elevated suicide risk is an important step toward the goal of "Zero Suicide." An immediate and unconventional challenge is that the identification of suicide attempts from medical claims contains substantial uncertainty: almost 20% of "suspected" suicide attempts are identified from diagnosis codes indicating external causes of injury and poisoning with undermined intent. It is thus of great interest to learn which of these undetermined events are more likely actual suicide attempts and how to properly utilize them in survival analysis with severe censoring. To tackle these interrelated problems, we develop an integrative Cox cure model with regularization to perform survival regression with uncertain events and a latent cure fraction. We apply the proposed approach to study the risk of subsequent suicide attempts after suicide-related hospitalization for the adolescent and young adult population, using medical claims data from Connecticut. The identified risk factors are highly interpretable; more intriguingly, our method distinguishes the risk factors that are most helpful in assessing either susceptibility or timing of subsequent attempts. The predicted statuses of the uncertain attempts are further investigated, leading to several new insights on suicide event identification.