  We use the correspondence between extremal Sasaki structures and weighted
extremal Kahler metrics defined on a regular quotient of a Sasaki manifold,
established by the first two authors, and Lahdili's theory of weighted
K-stability in order to define a suitable notion of (relative) weighted
K-stability for compact Sasaki manifolds of regular type. We show that the
(relative) weighted K-stability with respect to a maximal torus is a necessary
condition for the existence of a (possibly irregular) extremal Sasaki metric.
We also compare weighted K-stability to the K-stability of the corresponding
polarized affine cone (introduced by Collins-Szekelyhidi), and prove that they
agree on the class of test configurations we consider. As a byproduct, we
strengthen the obstruction to the existence of a scalar-flat Kahler cone metric
from the K-semistability to the K-stability on these test configurations. We
use our approach to give a characterization of the existence of a compatible
extremal Sasaki structure on a principal circle bundle over an admissible ruled
manifold, expressed in terms of the positivity of a single polynomial of one
variable over a given interval.
