  We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous
nonlinear Schr\"{o}dinger (INLS) equations. For $N\ge 3$ we treat the equation
with combined Hardy-Sobolev power-type nonlinearities $$
  -\Delta u+\lambda u=\mu|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u
\;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $\lambda\in\mathbb{R}$,
$\mu>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ and $2^*_{d}= 2(N-d)/(N-2)$
is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the
equation with critical exponential growth \begin{equation}\nonumber
\begin{aligned}
  &-\Delta u+\lambda u=|x|^{-b}f(u) \;\;\mbox{in}\;\; \mathbb{R}^2
\end{aligned} \end{equation} where the nonlinearity $f(s)$ behaves like
$\exp(s^2)$ as $s\to\infty$. We extend the existence results due to
Alves-Ji-Miyagaki (Calc. Var. 61, 2022) from $b =d= 0$ to the case $0 < b,d <
2$.
