We consider detecting and estimating breaks in heterogenous mean functions of high-dimensional functional time series which are allowed to be cross-sectionally correlated. A new test statistic combining the functional CUSUM statistic and power enhancement component is proposed with asymptotic null distribution comparable to the conventional CUSUM theory derived for a single functional time series. In particular, the extra power enhancement component enlarges the region where the proposed test has power, and results in stable power performance when breaks are sparse in the alternative hypothesis. Furthermore, we impose a latent group structure on the subjects with heterogenous break points and introduce an easy-to-implement clustering algorithm with an information criterion to consistently estimate the unknown group number and membership. The estimated group structure improves the convergence property of the break point estimate. Monte Carlo simulation studies and empirical applications show that the proposed estimation and testing techniques have satisfactory performance in finite samples.