For time-to-event outcomes, the bias introduced by covariate measurement error and the methods of analysis to address this bias have been well-studied. Less attention has been given to addressing errors in a failure time outcome, as well as to the setting where there are correlated errors in both the outcome and covariates - a common setting in medical applications that depend on electronic health records data. We develop a regression calibration type estimator for a general underlying error model that allows for errors in a failure time outcome that are possibly correlated with errors in covariates. In particular, we consider an adaption of two estimators developed to address covariate measurement error in the Cox model, including that of Prentice (Biometrika, 1982) and the risk set regression calibration method of Xie et al. (JRSSB, 2001), which recalibrates the error-prone covariates at multiple event times. Detailed numerical studies are presented to examine the performance of our estimators under varying scenarios of the error structure, strength of association, and level of censoring. We further illustrate our method with an application to real data.