  In this note, we prove $\mathcal{C}^{1,\gamma}$ regularity for solutions of
some fully nonlinear degenerate elliptic equations with "superlinear" and
"subquadratic " Hamiltonian terms. As an application, we complete the results
of \cite{BDL1} concerning the associated ergodic problem, proving, among other
facts, the uniqueness, up to constants, of the ergodic function.
