  In this paper, we derive estimates for scalar curvature type equations with
more singular right hand side. As an application, we prove Donaldson's
conjecture on the equivalence between geodesic stability and existence of cscK
when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the
properness of $K$-energy with respect to a suitably defined distance implies
the existence of cscK.
