  Unifying two notions of an action and coaction of a locally compact group on
a $C^*$-cor\-re\-spond\-ence we introduce a coaction $(\sigma,\delta)$ of a
Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. We show that
this coaction naturally induces a coaction $\zeta$ of $S$ on the associated
Cuntz-Pimsner algebra $\mathcal{O}_X$ under the weak $\delta$-invariancy for
the ideal $J_X$. When the Hopf $C^*$-algebra $S$ is defined by a well-behaved
multiplicative unitary, we construct a $C^*$-cor\-re\-spond\-ence
$(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ from
$(\sigma,\delta)$ and show that it has a representation on the reduced crossed
product $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ by the induced coaction
$\zeta$. This representation is used to prove an isomorphism between the
$C^*$-algebra $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ and the Cuntz-Pimsner
algebra $\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$ under the covariance
assumption which is guaranteed in particular if the ideal
$J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$ is generated
by the canonical image of $J_X$ in $M(A\rtimes_\delta\widehat{S})$ or the left
action on $X$ by $A$ is injective. Under this covariance assumption, our
results extend the isomorphism result known for actions of amenable groups to
arbitrary locally compact groups. The Cuntz-Pimsner covariance condition which
was assumed for the same isomorphism result concerning group coactions is shown
to be redundant.
