  We derive an a priori real Hessian estimate for solutions of a large family
of geometric fully non-linear elliptic equations on compact Hermitian
manifolds, which is independent of a lower bound for the right-hand side
function. This improves on the estimates of Sz\'ekelyhidi and additionally
applies to elliptic equations with a degenerate right-hand side. As an
application, we establish the optimal $C^{1,1}$ regularity of envelopes of
$(\theta, m)$-subharmonic functions on compact Hermitian manifolds.
