  We introduce the Haagerup property for twisted groupoid $C^*$-dynamical
systems in terms of naturally defined positive-definite operator-valued
multipliers. By developing a version of `the Haagerup trick' we prove that this
property is equivalent to the Haagerup property of the reduced crossed product
$C^*$-algebra with respect to the canonical conditional expectation $E$. This
extends a theorem of Dong and Ruan for discrete group actions, and implies that
a given Cartan inclusion of separable $C^*$-algebras has the Haagerup property
if and only if the associated Weyl groupoid has the Haagerup property in the
sense of Tu. We use the latter statement to prove that every separable
$C^*$-algebra which has the Haagerup property with respect to some Cartan
subalgebra satisfies the Universal Coefficient Theorem. This generalises a
recent result of Barlak and Li on the UCT for nuclear Cartan pairs.
