  Given a cocommutative Hopf algebra $\mathcal{H}$ over a commutative ring $K$
and a symmetric partial action of $\mathcal{H}$ on a $K$-algebra $A,$ we obtain
a first quadrant Grothendieck spectral sequence converging to the Hochschild
homology of the smash product $A \# \mathcal{H},$ involving the Hochschild
homology of $A$ and the partial homology of $\mathcal{H}.$ An analogous third
quadrant cohomological spectral sequence is also obtained. The definition of
the partial (co)homology of $\mathcal{H}$ under consideration is based on the
category of the partial representations of $\mathcal{H}.$ A specific partial
representation of $\mathcal{H}$ on a subalgebra $\mathcal{B}$ of the partial
``Hopf" algebra $\mathcal{H}_{par} $ is involved in the definition and we
construct a projective resolution of $\mathcal{B}.$
