Recurrent event data can be easily found in longitudinal studies such as clinical trials, reliability fields, and the social sciences; however, there are a few observations that disappear temporarily in sight during the follow-up and then suddenly reappear without notice like the Young Traffic Offenders Program(YTOP) data collected by Farmer et al. (2000). In this article we focused on inference for a cumulative mean function of the recurrent event data with these incomplete observation gaps. Defining a corresponding risk set would be easily accomplished if we know the exact intervals where the observation gaps occur. However, when they are incomplete (if their starting times are known but their terminating times are unknown) we need to estimate a distribution function for the terminating times of the observation gaps. To accomplish this, we treated them as interval-censored and then estimated their distribution using the EM algorithm proposed by Turnbull (1976). We proposed a nonparametric estimator for the cumulative mean function and also a nonparametric test to compare the cumulative mean functions of two groups. Through simulation we investigated the finite-sample performance of the proposed estimator and proposed test. Finally, we applied the proposed methods to YTOP data.