  We define the secondary Hochschild complex for an entwining structure over a
commutative $k$-algebra $B$. We show that this complex carries the structure of
a weak comp algebra. We obtain two distinct cup product structures for the
secondary cohomology groups. We also consider a subcomplex on which the two cup
products coincide and which satisfies the axioms for being a comp algebra. The
cohomology of this subcomplex then forms a Gerstenhaber algebra. We also
construct a bicomplex that controls the deformations of the entwining structure
over $B$.
