Let $\sum_{i=1}^{n}g(x_{i}) = G$. What is important is that I do't know the value of $g(x_{1}),...,g(x_{n})$, and I only know the value of their summation.
How I can compute the answer of the following summation based on $G$ $$\sum_{i=1}^{n}g(x_{i})e^{x_{i}}$$
Assuming you don't know either $g$, $x_i$, it is impossible. It is possible for example that $g(x) = x$.
Then $G = 2$ for both $x = (2, 0)$ and $x = (1, 1)$. So you can't distinguish between those two cases.
But one has answer $2e^2$ while the other has answer $2e$.