  We consider an arbitrary deformation of the Gaussian matrix model
parameterized by Miwa variables $z_a$. One can look at it as a mixture of the
Gaussian and logarithmic (Selberg) potentials, which are both superintegrable.
The mixture is not, still one can find an explicit expression for an arbitrary
Schur average as a linear transform of a {\it finite degree} polynomial made
from the values of skew Schur functions at the Gaussian locus
$p_k=\delta_{k,2}$. This linear operation includes multiplication with an
exponential $ e^{z_a^2/2}$ and a kind of Borel transform of the resulting
product, which we call multiple and enhanced. The existence of such remarkable
formulas appears intimately related to the theory of auxiliary $K$-polynomials,
which appeared in {\it bilinear} superintegrable correlators at the Gaussian
point (strict superintegrability). We also consider in the very detail the
generating function of correlators $<(\Tr X)^k>$ in this model, and discuss its
integrable determinant representation. At last, we describe deformation of all
results to the Gaussian $\beta$-ensemble.
