  We study the following version of Hardy-type inequality on a domain $\Omega$
in a Riemannian manifold $(M,g)$:
  $$
  \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g
  \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla
\rho|_g^p}{|\rho|^p}\rho^\alpha dV_g
  +\int{\Omega} V|u|^p\rho^\alpha dV_g,
  \quad \forall\ u\in C_c^\infty (\Omega).
  $$
  We provide sufficient conditions on $p, \alpha, \beta,\rho$ and $V$ for which
the above inequality holds. This generalizes earlier well-known works on Hardy
inequalities on Riemannian manifolds. The functional setup covers a wide
variety of particular cases, which are discussed briefly: for example,
$\mathbb{R}^N$ with $p<N$, $\mathbb{R}^N\setminus \{0\}$ with $p\geq N$,
$\mathbb{H}^N$, etc.
