Recurrent event data arise frequently in many longitudinal follow-up studies. Hence, evaluating covariate effects on the rates of occurrence of such events is commonly of interest. Examples include repeated hospitalizations, recurrent infections of HIV, and tumor recurrences. In this article, we consider semiparametric regression methods for the occurrence rate function of recurrent events when the covariates may be measured with errors. In contrast to the existing works, in our case the conventional assumption of independent censoring is violated since the recurrent event process is interrupted by some correlated events, which is called informative drop-out. Further, some covariates may be measured with errors. To accommodate for both informative censoring and measurement error, the occurrence of recurrent events is modelled through an unspecified frailty distribution and accompanied with a classical measurement error model. We propose two corrected approaches based on different ideas, and we show that they are numerically identical when estimating the regression parameters. The asymptotic properties of the proposed estimators are established, and the finite sample performance is examined via simulations. The proposed methods are applied to the Nutritional Prevention of Cancer trial for assessing the effect of the plasma selenium treatment on the recurrence of squamous cell carcinoma.