Considering the monoidal category $\mathcal{ C}$ obtained as modules over a Hopf algebra $H$ in a rigid braided category $\mathcal{ B}$, we prove decomposition results for the Hochschild and cyclic homology categories $HH(\mathcal{ C})$ and $HC(\mathcal{ C})$ of $\mathcal{ C}$. This is accomplished by defining a notion of a (stable) anti-Yetter-Drinfeld module with coefficients in a (stable) braided module over $\mathcal{ B}$. When the stable braided module is $HH(\mathcal{ B})$, we recover $HH(\mathcal{ C})$ and $HC(\mathcal{ C})$. The decomposition of $HC(\mathcal{ C})$ now follows from that of $HH(\mathcal{ B})$.
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