  All unitary perturbations of a given unitary operator $U$ by finite rank $d$
operators with fixed range can be parametrized by $(d\times d)$ unitary
matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations,
where the Aleksandrov--Clark family of unitary perturbations is parametrized by
the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$.
  For a purely contractive $\Gamma$ the resulting perturbed operator $T_\Gamma$
is a contraction (a completely non-unitary contraction under the natural
assumption about cyclicity of the range), so they admit the functional model.
  In this paper we investigate the Clark operator, i.e. a unitary operator that
intertwines $T_\Gamma$ (presented in the spectral representation of the
non-perturbed operator $U$) and its model. We make no assumptions on the
spectral type of the unitary operator $U$; absolutely continuous spectrum may
be present.
  We find a representation of the adjoint Clark operator in the coordinate free
Nikolski--Vasyunin functional model. This representation features a special
version of the vector-valued Cauchy integral operator. Regularization of this
singular integral operator yield representations of the adjoint Clark operator
in the Sz.-Nagy--Foias transcription. In the special case of inner
characteristic functions (purely singular spectral measure of $U$) this
representation gives what can be considered as a natural generalization of the
normalized Cauchy transform (which is a prominent object in the Clark theory
for rank one case) to the vector-valued settings.
