  Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the
twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the
partial projective $\sigma$-representations of $G$ into algebras over a filed
$\kappa.$ Using the relation between partial projective representations and
twisted partial actions we endow $\kappa_{par}^\sigma G$ with the structure of
a crossed product by a twisted partial action of $G$ on a commutative
subalgebra of $\kappa_{par}^{\sigma} G.$ Then, we use twisted partial group
algebras to obtain a first quadrant Grothendieck spectral sequence converging
to the Hochschild homology of the crossed product $A\ast_{\Theta} G,$ involving
the Hochschild homology of $A$ and the partial homology of $G,$ where
${\Theta}$ is a unital twisted partial action of $G$ on a $\kappa$-algebra $A$
with a $\kappa $-based twist. An analogous third quadrant cohomological
spectral sequence is also obtained.
