This article concerns item nonresponse adjustment for two-stage cluster samples, where nonresponse depends on covariates and underlying cluster characteristics, or on covariates and the missing outcome. In these circumstances, standard weighting and imputation adjustments are liable to be biased. To obtain consistent estimates, we propose extensions of the standard random-effects model for clustered data to model these two types of missing data mechanisms. These new methods are compared with existing approaches by simulation studies, and illustrated on data on household income from the Behavioral Risk Factor Surveillance System.