Let $$\Omega \subset {\mathbb R}^2$$ be a bounded domain with boundary of class $$C^2$$ . The purpose of this paper is to study the existence of positive solution for the following inhomogeneous singular Neumann problem: $$\begin{aligned} (P_{\mu ,\lambda })\qquad \left\{ \begin{array}{ll} &{} - \Delta u+u = \mu u^{-\delta }+h(x,u)e^{u^\alpha }, \quad u>0 \quad \text {in } \Omega , \\ &{}\frac{\partial u}{\partial \nu }= \lambda \psi u^q \quad \text {on }\partial \Omega . \end{array}\right. \end{aligned}$$ where $$\mu ,\lambda >0,$$ $$0<\delta <3,$$ $$1\le \alpha \le 2,$$ $$0\le q<1,$$ and $$\psi $$ is a non-negative Holder continuous function on $$\overline{\Omega }.$$ Here, h(x, u) is a $$C^{1}(\overline{\Omega }\times \mathbb {R})$$ having superlinear growth at infinity. Using variational methods, we show that there exists a region $$\mathcal {R}\subset \{(\mu ,\lambda ):\;\mu ,\lambda >0\}$$ bounded by the graph of a map $$\Lambda $$ , such that $$(P_{\mu ,\lambda })$$ admits at least two solutions for all $$(\mu ,\lambda ) \in \mathcal {R},$$ at least one solution for $$(\mu ,\lambda )\in \partial \mathcal {R}$$ and no solution for all $$(\mu ,\lambda )$$ outside $$\overline{\mathcal {R}}.$$