In a graph where vertices have preferences over their neighbors, a matching is called popular if it does not lose a head-to-head election against any other matching when the vertices vote between the matchings. Popular matchings can be seen as an intermediate category between stable matchings and maximum-size matchings. In this article, we aim to maximize the utility of a matching that is popular but admits only a few blocking edges. We observe that, for general graphs, finding a popular matching with at most one blocking edge is already NP -complete. For bipartite instances, we study the problem of finding a maximum-utility popular matching with a bound on the number (or, more generally, the cost) of blocking edges applying a multivariate approach. We show classical and parameterized hardness results for severely restricted instances. By contrast, we design an algorithm for instances where preferences on one side admit a master list and show that this algorithm is roughly optimal.