  In this paper we study a Ricci-Hessian type manifold
$(\Bbb{M},g,\varphi,f,\lambda)$ which is closely related to the construction of
almost Ricci soliton realized as a warped product. We classify certain classes
of the Ricci-Hessian type manifolds and derive some implications for almost
Ricci solitons and generalized $m$--quasi-Einstein manifolds. We consider two
complementary cases: $\nabla f$ and $\nabla\varphi$ are linearly independent in
$C^\infty(\Bbb{M})$--module $\mathfrak{X}(\Bbb{M})$; and $\nabla
f=h\nabla\varphi$ for a smooth function $h$ on $\Bbb{M}$. In the first case we
show that the vector field $\nabla\lambda$ belongs to the
$C^\infty(\Bbb{M})$--module generated by $\nabla f$ and $\nabla\varphi$, while
in the second case, under additional hypothesis, the manifold is, around any
regular point of $f$, locally isometric to a warped product.
