In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:

$$\{\begin{array}{rl}& -\left(a+b{\int }_{{\mathbb{R}}^N}{\left|\mathrm{\nabla }u\right|}^{2}dx\right)\mathrm{\Delta }u+\lambda u=|u{|}^{p-2}u+h(x){\left|u\right|}^{q-2}u,\text{in}{\mathbb{R}}^N,\\ & {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=c,u\in H^1({\mathbb{R}}^N),\end{array}$$

where $1\le N\le 3,a,b,c>0,1\le q<2$, $\lambda \in \mathbb{R}$. We treated three cases:

(i) When $2<p<2+\frac{4}{N},h(x)\ge 0$, we obtained the existence of a global constraint minimizer.

(ii) When $2+\frac{8}{N}<p<2^{\ast },h(x)\ge 0$, we proved the existence of a mountain pass solution.

(iii) When $2+\frac{8}{N}<p<2^{\ast },h(x)\le 0$, we established the existence of a bound state solution.