Your question is too general because you don't specify $g(z)$ and $C$. Something can be done for more concrete cases e.g. $g(z):=z^n,\, n \in \mathbb{N}, C:=\{z:|z|=2\}$ with Maple 2019.1:

    int(2*eval(exp(1/z) + exp(2/(z - 1)), z = 2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi);

>$6\,\pi$

    int(2*eval(z^4*(exp(1/z) + exp(2/(z - 1))), z = 2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi);

> ${\frac {2513\,\pi}{60}}$

Unfortunately, a more general case

    int(2*eval(z^n*(exp(1/z) + exp(1/(z-1))),z=2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi)assuming n::posint;

fails.