In this paper, we discuss the statistical inference of interval-censored recurrence event data under an informative observation process. We establish an additive semiparametric mean model for the recurrence event process. Since the observation process may contain relevant information about potential underlying recurrence event processes, which leads to confounding bias, therefore, we introduced a propensity score into the additive semiparametric mean model to adjust for confounding bias, which possibly exists. Furthermore, the estimation equations were used to estimate the parameters of the covariate effects, and the asymptotic normality of the estimator under large samples is proven. Through simulation studies, we illustrated that the proposed method works well, and it was applied to the analysis of bladder cancer data.