In this paper, we investigate the following non-autonomous Kirchhoff equation: \(\begin{aligned} & -\Big (a+b\int _{{\mathbb {R}}^{4}}|\nabla u|^{2}{\textrm{d}}x\Big )\Delta u+V(\epsilon x)u\\ & \quad =\lambda u+\mu (I_{\alpha }*|u|^{p})|u|^{p-2}u+|u|^{2}u,\quad x\in {\mathbb {R}}^{4} \end{aligned}\) with prescribed mass \(\int _{{\mathbb {R}}^{4}}|u|^{2}{\textrm{d}}x=c^{2},\) where \(a,b,c,\mu >0,\) \(2<\alpha <4,\) \(2<p<\frac{6+\alpha }{4},\) and \(\epsilon >0\) is a small parameter. Here, \(\lambda \in {\mathbb {R}}\) appears as a Lagrange multiplier, while the potential V is a continuous nonnegative function satisfying specified global assumptions. Our principal results are divided into two parts depending on the conditions imposed on the potential V. First, we establish the existence of a normalized solution to this problem when V is a nonnegative constant within an appropriate range. Moreover, by employing the truncated argument and minimization techniques, we show that the number of normalized solutions is not less than the numbers of global minimum points of V.
