  Asymptotic expansion of the distribution of a perturbation $Z_n$ of a
Skorohod integral jointly with a reference variable $X_n$ is derived. We
introduce a second-order interpolation formula in frequency domain to expand a
characteristic functional and combine it with the scheme developed in the
martingale expansion. The second-order interpolation and Fourier inversion give
asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable
functions $f$ and also measurable functions $f$. In the latter case, the
interpolation method connects the two non-degeneracies of variables for finite
$n$ and $\infty$. Random symbols are used for expressing the asymptotic
expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are
introduced in this paper. We identify these random symbols for a certain
quadratic form of a fractional Brownian motion and for a quadratic from of a
fractional Brownian motion with random weights. For a quadratic form of a
Brownian motion with random weights, we observe that our formula reproduces the
formula originally obtained by the martingale expansion.
