Suppose that you have two random variables $x\sim F_x$ and $y\sim F_y$ where $F_x,F_y$ are probability distributions. Then generally $w_xx+w_yy$ is not from $w_xF_x+w_yF_y$, even if when $F_x,F_y$ are normal distributions.

It is true that the characteristic function of the weighted sum is the sum of characteristic functions of these distributions: $\psi_{w_xx+w_yy}=w_x\psi_x+w_y\psi_y$. You can obtain the distribution $F_{w_xx+w_yy}$ from its characteristic function, of course.

Finally, the whole idea of the Gaussian mixture is to model unknown distributions, such as those with fat tails in finance.

Suppose that you have two random variables $x\sim F_x$ and $y\sim F_y$ where $F_x,F_y$ are probability distributions. Then generally $w_xx+w_yy$ is not from $w_xF_x+w_yF_y$, even if when $F_x,F_y$ are normal distributions.

The characteristic function of the weighted sum is the product of characteristic functions of these distributions: $\psi_{w_xx+w_yy}(t)=\psi_x(w_xt)\times \psi_y(w_yt)$. You can obtain the distribution $F_{w_xx+w_yy}$ from its characteristic function, of course.

Finally, the whole idea of the Gaussian mixture is to model unknown distributions, such as those with fat tails in finance.

Now, what is meant by $w_x\mathcal N+w_y\mathcal N$? In case of a mixture it is the weighted average density (PDF).