<p>In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{aligned} &amp;-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u = |u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &amp;\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx = c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $\end{document} </tex-math></disp-formula></p><p>where $ 1\le N\le 3, a, b, c &gt; 0, 1\leq q &lt; 2 $, $ \lambda \in \mathbb{R} $. We treated three cases:</p><p>(i) When $ 2 &lt; p &lt; 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.</p><p>(ii) When $ 2+\frac{8}{N} &lt; p &lt; 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.</p><p>(iii) When $ 2+\frac{8}{N} &lt; p &lt; 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.</p>