When analyzing data from randomized clinical trials, covariate adjustment can be used to account for chance imbalance in baseline covariates and to enhance the precision of the treatment effect estimate. A practical barrier to implementing covariate adjustment is the presence of missing data. We investigate the implications of the missing data mechanism on the estimation of the average treatment effect in randomized clinical trials. We propose a weighting approach that combines inverse probability weighting for adjusting missing outcomes and overlap weighting for covariate adjustment. We conduct simulation studies to examine the finite sample performance of the proposed methods. We find that the proposed adjustment methods generally improve the precision of treatment effect estimates, irrespective of the imputation methods, when the adjusted covariate is associated with the outcome.