  Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on
$\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a
generalized mean field equation \begin{align*}
  -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma
he^u}-\dfrac{1}{\mathrm{Area}\left(\Sigma\right)}\right)+\alpha\left(u-\fint_{\Sigma}u\right),
\end{align*} where $\Delta$ denotes the Laplace-Beltrami operator. We first
derive a uniform bound for solutions when $\rho\in (8k\pi, 8(k+1)\pi)$ for some
non-negative integer number $k\in \mathbb{N}$ and
$\alpha\notin\mathrm{Spec}\left(-\Delta\right)\setminus\set{0}$. Then we obtain
existence results for $\alpha<\lambda_1\left(\Sigma\right)$ by using the
Leray-Schauder degree theory and the minimax method, where
$\lambda_1\left(\Sigma\right)$ is the first positive eigenvalue for $-\Delta$.
