  This paper establishes Carleson embeddings of M{\"u}ntz spaces
$M^q_{\Lambda}$ into weighted Lebesgue spaces $L^p(\mathrm{d}\mu)$, where $\mu$
is a Borel regular measure on $[0,1]$ satisfying $\mu([1-\varepsilon])\lesssim
\varepsilon^{\beta}$. In the case $\beta \geqslant 1$ we show that such
measures are exactly the ones for which Carleson embeddings
$L^{\frac{p}{\beta}} \hookrightarrow L^p(\mathrm{d}\mu)$ hold. The case $\beta
\in (0,1)$ is more intricate but we characterize such measures $\mu$ in terms
of a summability condition on their moments. Our proof relies on a
generalization of $L^p$ estimates {\`a} la Gurariy-Macaev in the weighted $L^p$
spaces setting, which we think can be of interest in other contexts.
