On the hardness of addition when the elements of a field is represented by the powers of generator and possible any existant scheme
The term used for the function $Z(\cdot)$ defined by
$$
g^{x}+g^{y}=g^{x}(1+g^{y-x})=g^x g^{Z(y-x)},
$$
in coding theory literature is Zech's logarithm. So for your formulation $$z=x+Z(y-x)=y+Z(x-y).$$
It is no more than the DL of $(1+g^{x-y})$ and is believed to be as hard as DL.
In coding theory it is normally considered for even characteristic groups $G=GF(2^m)^\ast$ in which case it is actually just the DL composed with a permutation $\pi$ of order 2 in the additive exponent group $\mathbb{Z}_{2^m-1}.$
I seem to recall Odlyzko mentioning it in a survey on DL in early 1990's. In any case, I don't know of it being directly used in cryptographic schemes, but others might.