Given a metric space, the (k, z)-clustering problem consists of finding k centers such that the sum of the of distances raised to the power z of every point to its closest center is minimized. This encapsulates the famous k-median (z = 1) and k-means (z = 2) clustering problems. Designing small-space sketches of the data that approximately preserves the cost of the solutions, also known as coresets, has been an important research direction over the last 15 years. In this paper, we present a new, simple coreset framework that simultaneously improves upon the best known bounds for a large variety of settings, ranging from Euclidean space, doubling metric, minor-free metric, and the general metric cases: with Γ = min(ε −2 +ε −z , kε −2)polylog(ε −1), this framework constructs coreset with size • O (Γ • k(d + log k)) in doubling metrics, improving upon the recent breakthrough of [Huang, Jiang, Li, Wu, FOCS' 18], who presented a coreset with size O(k 3 d/ε 2). • O(Γ•k•min(d, ε −2 log k)) in d-dimensional Euclidean space, improving on the recent results of [Huang, Vishnoi, STOC' 20], who presented a coreset of size O(k log k • ε −2z • min(d, ε −2 log k)). • O(Γ • k(t + log k)) for graphs with treewidth t, improving on [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML'20], who presented a coreset of size O(k 2 t/ε 2) for z = 1. • O Γ • k log 2 k + log k ε 4