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In many applications involving regression analysis, explanatory variables (or covariates) may be imprecisely measured or may contain missing values. Although there exists a vast literature on measurement error modeling to account for errors-in-variables, and on missing data methodology to handle missingness, very few methods have been developed to simultaneously address both. In this paper, we consider likelihood-based multiple imputation to handle missing data, and combine this with two well-known functional measurement error methods: simulation-extrapolation and corrected score. This unified approach has several appealing characteristics: the model fitting procedure is easy to understand and off-the-shelf software can be incorporated into the modeling framework; no calibration data or a validation subset is required in the model fitting procedure; and the missing data component of the proposed approach is likelihood-based which allows standard likelihood machinery. We demonstrate our methods on simulated datasets and apply them to daily ozone pollution measurements in Los Angeles where observed covariates consist of missing data and imprecise measurements. We conclude that the proposed methods substantially reduce bias and mean squared errors in regression coefficients, in comparison to methods that ignore either measurement error or missingness in covariates. [ABSTRACT FROM AUTHOR]