This work is motivated by a nephrology study in Taiwan, where, after shunt implantation, dialysis patients may experience one of the two types, acute and non-acute, of shunt thrombosis, and each of them may alternatively recur in a patient. In this work, treating the two types of shunt thrombosis as competing risks, we assess covariate effects on the cumulative incidence probability function, or subdistribution, of gap times to the occurrences of acute shunt thrombosis. To accommodate potentially time-varying covariate effects, we extend a varying-coefficient subdistribution regression model to recurrent event analysis and propose associated estimation procedures. The inverse probability of censoring weighting technique is employed to ensure consistent estimation of the regression parameter. Asymptotic distributional theory is derived for the proposed estimator. Simulation results confirm that the proposed estimator performs well in finite samples. Application of the proposed analysis to the shunt thrombosis data reveals that dialysis patients with graft shunts and hypertension are associated with significantly increased incidence of acute shunt thrombosis.