A robust likelihood approach for the analysis of overdispersed correlated count data that takes into account cluster varying covariates is proposed. We emphasise two characteristics of the proposed method: That the correlation structure satises the constraints on the second moments and that the estimation of the correlation structure guarantees consistent estimates of the regression coecients. In addition we extend the mean specication to include within- and between-cluster eects. The method is illustrated through the analysis of data from two studies. In the rst study, cross-sectional count data from a randomised controlled trial are analysed to evaluate the ecacy of a communication skills training programme. The second study involves longitudinal count data which represent counts of damaged hand joints in patients with psoriatic arthritis. Motivated by this study, we generalize our model to accommodate for a subpopulation of patients who are not susceptible to the development of damaged hand joints.