  We study coresets for clustering with capacity and fairness constraints. Our
main result is a near-linear time algorithm to construct
$\tilde{O}(k^2\varepsilon^{-2z-2})$-sized $\varepsilon$-coresets for
capacitated $(k,z)$-clustering which improves a recent
$\tilde{O}(k^3\varepsilon^{-3z-2})$ bound by [BCAJ+22, HJLW23]. As a corollary,
we also save a factor of $k \varepsilon^{-z}$ on the coreset size for fair
$(k,z)$-clustering compared to them.
  We fundamentally improve the hierarchical uniform sampling framework of
[BCAJ+22] by adaptively selecting sample size on each ring instance,
proportional to its clustering cost to an optimal solution. Our analysis relies
on a key geometric observation that reduces the number of total ``effective
centers" from [BCAJ+22]'s $\tilde{O}(k^2\varepsilon^{-z})$ to merely $O(k\log
\varepsilon^{-1})$ by being able to ``ignore'' all center points that are too
far or too close to the ring center.
