Which is the total energy of the product of two discrete energy signals
You don't give enough information for an expression of the combined energy in your expression.
$$\begin{eqnarray}
&amp; &amp;\sum_{i=0}^N \left| x_s(i)\cdot\left( 1+K\,w(i) \right) \right|^2 \\
&amp;=&amp;\sum_{i=0}^N \left( x_s(i)\cdot\left( 1+K\,w(i) \right) \right)^\star \times\left( x_s(i) \cdot\left( 1+K\,w(i) \right) \right)\\
&amp;=&amp;\sum_{i=0}^N \left( x_s(i)^\star\,x_s(i) + x_s(i)^\star\,K^\star\,w(i)^\star x_s(i)\,K\,w(i) + x_s(i)^\star x_s(i)\,K\,w(i)+x_s(i)\,x_s(i)^\star\,K^\star w(i)^\star\right)\\
&amp;=&amp;\sum_{i=0}^N \left( |x_s(i)|^2 \times \left(1+2\,\Re(K\,w(i))+|K|^2\,|w(i)|^2 \right) \right)\\
&amp;=&amp;E_s+\sum_{i=0}^N|x_s(i)|^2\left(2\,\Re(K\,w(i))+|K|^2\,|w(i)|^2 \right)
\end{eqnarray}$$

This means you need to know the correlation of the square magnitude of $x_s$ with the real part of $w$ and the square magnitude of $w$. The correlation of just $x_s$ and $w$ does not play any role.