  In this paper, we study the generalized mean-field stochastic control problem
when the usual stochastic maximum principle (SMP) is not applicable due to the
singularity of the Hamiltonian function. In this case, we derive a second order
SMP. We introduce the adjoint process by the generalized mean-field backward
stochastic differential equation. The keys in the proofs are the expansion of
the cost functional in terms of a perturbation parameter, and the use of the
range theorem for vector-valued measures.
