In this letter, we study the problem of fault detection and mitigation in networks where the measurements satisfy Kirchhoff's voltage law. First, we characterise the class of faults appearing as an additive fault vector (injected by a malicious adversary or due to equipment failures) that can be detected by taking into account the topology of the network. Second, we consider the problem of estimating the fault vector via tools from compressive sensing. Moreover, we comment on the applicability of the developed methods to the case where the measurements satisfy Kirchhoff's current law. The proposed methods are validated via numerical examples with application to time synchronization networks.