A Poisson-gamma model is introduced to account for between-subjects heterogeneity and within-subjects serial correlation occurring in longitudinal count data. The model extends the usual time-constant shared frailty approach to allow time-varying serially correlated gamma frailty whilst retaining standard marginal assumptions. A composite likelihood approach to estimation and testing for serial correlation is proposed. The work is motivated by a clinical trial on patient-controlled analgesia where the number of analgesic doses taken by hospital patients in successive time intervals following abdominal surgery is recorded.