  Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix an
integer $m$ such that $1\leq m\leq n$. We reformulate most relative
pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the
Hessian setting. As an application, we use a slightly different method and give
an characterization of finite energy range of the Hessian operator, which
cannot be directly reformulated by \cite{DNL23}.
  Given a model potential $\phi$, we also study degenerate complex Hessian
equations of the form $(\omega+dd^c
\varphi)^m\wedge\omega^{n-m}=F(x,\varphi)\omega^n$. Under some natrual
conditions on $F$, we prove that the solution of this type equation has the
same singularity type as $\phi$.
