Implication of $E(XY)\neq 0$ Consider two real-valued scalar random variables $X,Y$ such that
$$
(1) \quad E(XY)\neq 0,\quad E(X)= 0, \quad E(Y)= 0
$$
Let $g: \mathbb{R}\rightarrow \mathbb{R}$ be a function. Under which conditions on $g$, (1) implies
$$
E(Xg(Y))\neq 0 \quad ?
$$
PS: The comments below gives an insightful answer for the case where $E(XY)\neq 0,  E(X)\neq 0,  E(Y)\neq  0$

As the comments pointed out, the degree of generality sought by the OP is quixotic.
Let $(X,Y)$ have a joint symmetric bivariate distribution in the sense that
$$(X, Y) \sim_d (-X, -Y).$$
This already implies $E(X) = E(Y) = 0$, and we assume also that $E(XY) \neq 0$.
Let $g(z) = z^k$, $k$ is an even number. Then (see https://stats.stackexchange.com/a/352218/28746)
$$E[Xg(Y)] = 0,\;\;\; \forall k.$$
So, no, eq. (1) of the OP cannot imply that this coproduct will have non-zero expected value for any $g$ (as the OP hoped for in a comment), since for this class of distributions and for this $g$ this expected value becomes zero.
