  We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex
$\chH$ over free associative algebra or path algebra of a quiver. The $\chH$
complex is introduced as an appropriate tool for the definition of
pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in
case of free algebras (path algebras), concentrated in degree zero. It serves
as a main ingredient for the formality proof. For any smooth algebra we choose
a small qiso subcomplex in the higher cyclic Hochschild complex, which gives
rise to a calculus of highly noncommutative monomials, we call them
$\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially
described in terms of $\xi\delta$-monomials. This subcomplex and a basis of
$\xi\delta$-monomials in combination with arguments from Groebner bases theory
serves for the cohomology calculations of the higher cyclic Hochschild complex.
The language of $\xi\delta$-monomials in particular allows an interpretation of
pre-Calabi-Yau structure as a noncommutative Poisson structure.
