Many medical studies yield data on recurrent clinical events from populations which consist of a proportion of cured patients in the presence of those who experience the event at several times (uncured). A frailty mixture cure model has recently been postulated for such data, with an assumption that the random subject effect (frailty) of each uncured patient is constant across successive gap times between recurrent events. We propose two new models in a more general setting, assuming a multivariate time-varying frailty with an AR(1) correlation structure for each uncured patient and addressing multilevel recurrent event data originated from multi-institutional (multi-centre) clinical trials, using extra random effect terms to adjust for institution effect and treatment-by-institution interaction. To solve the difficulties in parameter estimation due to these highly complex correlation structures, we develop an efficient estimation procedure via an EM-type algorithm based on residual maximum likelihood (REML) through the generalised linear mixed model (GLMM) methodology. Simulation studies are presented to assess the performances of the models. Data sets from a colorectal cancer study and rhDNase multi-institutional clinical trial were analyzed to exemplify the proposed models. The results demonstrate a large positive AR(1) correlation among frailties across successive gap times, indicating a constant frailty may not be realistic in some situations. Comparisons of findings with existing frailty models are discussed.