  In this paper, we study the existence and non-existence of normalized
solutions to the lower critical Choquard equation with a local perturbation
\begin{equation*} \begin{cases} -\Delta u+\lambda u=\gamma
(I_{\alpha}\ast|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u+\mu
|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=c^2,
\end{cases} \end{equation*} where $\gamma, \mu, c>0$, $2<q\leq 2+\frac{4}{N}$,
and $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange
multiplier. The results of this paper about this equation answer some questions
proposed by Yao, Chen, R\v{a}dulescu and Sun [Siam J. Math. Anal., 54(3)
(2022), 3696-3723]. Moreover, based on the results obtained, we study the
multiplicity of normalized solutions to the non-autonomous Choquard equation
\begin{equation*} \begin{cases} -\Delta u+\lambda u=(I_\alpha\ast [h(\epsilon
x)|u|^{\frac{N+\alpha}{N}}])h(\epsilon
x)|u|^{\frac{N+\alpha}{N}-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^N, \\
\int_{\mathbb{R}^N}|u|^2dx=c^2, \end{cases} \end{equation*} where $\epsilon>0$,
$2<q<2+\frac{4}{N}$, and $h$ is a positive and continuous function. It is
proved that the numbers of normalized solutions are at least the numbers of
global maximum points of $h$ when $\epsilon$ is small enough.
