The paper deals with the existence of normalized solutions for the following planar Schrödinger-Poisson system with \(L^2\) -constraint: \({\left\{ \begin{array}{ll}-\Delta u+\lambda u+V(x)u+\mu \phi u= \left( e^{u^2}-1-u^2\right) u+\gamma |u|^{p-2} u, & x \in \mathbb {R}^2, \\ \Delta \phi =2 \pi u^2, & x \in \mathbb {R}^2,\\ \int _{\mathbb {R}^2} u^2 \textrm{d} x=m^{2}, \end{array}\right. }\) where \(m,\mu ,\gamma >0\) , \(p \in (2,4]\) , \( \lambda \in \mathbb {R}\) is a Lagrange multiplier, \(V \in C^{2}\left( \mathbb {R}^2\backslash \{0\},(0, \infty )\right) \) is a potential function. It is worth noting that the term \((e^{u^2}-1-u^2) u\) exhibits critical exponential growth characteristic. Besides this, we also need to consider the perturbation term \(\gamma |u|^{p-2} u\) focusing on the case \(p\in (2,4]\) and linear term V(x)u. By employing a range of variational methods to obtain the ground state solution of the equation, we also leverage the Moser function to establish an upper bound for the mountain pass level \(C_{\gamma , \mu }\) , which restores compactness. This allows us to further derive the mountain pass-type solution. Moreover, we further analyze the asymptotic behavior of these solutions.
