We generalize the naïve estimator of a Poisson regression model with measurement errors as discussed in Kukush et al. [1]. The explanatory variable is not always normally distributed as they assume. In this study, we assume that the explanatory variable and measurement error are not limited to a normal distribution. We clarify the requirements for the existence of the naïve estimator and derive its asymptotic bias and asymptotic mean squared error (MSE). In addition, we propose a consistent estimator of the true parameter by correcting the bias of the naïve estimator. As illustrative examples, we present simulation studies that compare the performance of the naïve estimator and new estimator for a Gamma explanatory variable with a normal error or a Gamma error.