This paper considers a problem of output agreement in heterogeneous networks with dynamics on the nodes as well as on the edges. The control and disturbance signals entering the nodal dynamics are “unmatched” meaning that some nodes are only subject to disturbances and not to the actuating signals. To further enrich our model and motivated by synchronization problems in physical networks, we accommodate (solvable) algebraic constraints resulting in a fairly general and heterogeneous network. It is shown that appropriate dynamic feedback controllers achieve output agreement on a desired vector, in the presence of physical coupling and despite the influence of constant as well as time-varying disturbances. Furthermore, we address the case of an optimal steady-state deployment of the control effort over the network by suitable distributed controllers. As a case study, the proposed results are applied to a heterogeneous microgrid.