  In this paper we study 4-dimensional $(m,\rho)$-quasi-Einstein manifolds with
harmonic Weyl curvature when $m\notin\{0,\pm1,-2,\pm\infty\}$ and
$\rho\notin\{\frac{1}{4},\frac{1}{6}\}$. We prove that a non-trivial
$(m,\rho)$-quasi-Einstein metric $g$ (not necessarily complete) is locally
isometric to one of the followings: (i) $\mathcal{B}^2_\frac{R}{2(m+2)}\times
\mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$ where $\mathcal{B}^2_\frac{R}{2(m+2)}$ is a
northern hemisphere in the 2-dimensional sphere
$\mathbb{S}^2_\frac{R}{2(m+2)}$, $\mathbb{N}_\delta$ is the 2-dimensional
Riemannian manifold with constant curvature $\delta$ and $R$ is the constant
scalar curvature of $g$, (ii)
$\mathcal{D}^2_\frac{R}{2(m+2)}\times\mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$ where
$\mathcal{D}^2_\frac{R}{2(m+2)}$ is one half (cut by a hyperbolic line) of the
hyperbolic plane $\mathbb{H}^2_\frac{R}{2(m+2)}$, (iii)
$\mathbb{H}^2_\frac{R}{2(m+2)}\times\mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$, (iv) a
certain singular metric with $\rho=0$, (vi) a locally conformally flat metric.
By applying this local classification, we obtain a classification of complete
$(m,\rho)$-quasi-Einstein manifolds under the harmonic Weyl curvature
condition. Our result can be viewed as a local classification of gradient
Einstein-type manifolds. One corollary of our result is the classification of
$(\lambda,4+m)$-Einstein manifolds which can be viewed as
$(m,0)$-quasi-Einstein manifolds.
