In signal averaging, will the noise be reduced if the noise is not Gaussian?
Consider two zero mean signals $x(n)$ and $y(n)$. Let $z(n)=x(n)+y(n)$. In the following $E()$ denotes the expectation.
$$E(z) = E(x+y)= E(x)+E(y)= 0$$
I've dropped the time index $n$ for convenience. So $z(n)$ has zero mean. Now let's look at the variance. Since $z(n)$ is zero mean - I'll just use the correlation.
$$E[z^2] =E[(x+y)(x+y)]=E[x^2+2xy+y^2]=E(x^2)+2E(xy)+E(y^2) $$

If the correlation between $x(n)$ and $y(n)$ is zero, then $E(xy)=0$, then the variances add. Usually this is done by stating the $x(n)$ and $(y)$ are independent which implies uncorrelated.

In the general case $E(xy)\neq 0$, so the resulting variance could increase or decrease depending on the sign of the correlation.

Note - stating that $x(n)$ and $y(n)$ are white can be a little unclear. Usually saying $x(n)$ is white implies it is uncorrelated with itself - it does not state how it is correlated to another noise process.

The above analysis is not dependent on the type of noise (Gaussian or not) and it does not make any assumptions on the autocorrelation functions of $x(n)$ and $y(n)$.