Our assumptions are: $\left(X_t\right)_{t\geqslant 0}$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly varying function. Let $Y_t:=G\left(X_t\right)$ with a real function $G$ such that $\mathbb{E}\left(G\left(X_1\right)\right)=0$ and $\mathbb{E}\left(G\left(X_1\right)^2\right)<\infty$. We know that for the ACF holds
$$\gamma_Y (k)\sim\frac{J(m)^2}{m!}L_\gamma^m(k)k^{m(2d-1)}$$
for $k\rightarrow \infty$, where $m$ is the Hermite rank of $G$, $J(m):=\mathbb{E}\left[G(X)H_m(X)\right]$ and $H_m$ is the $m$-th Hermite polynomial. What can we now say about the asymptotic behaviour of the sum of covariances
$$\sum_{j=1}^{\ell}\sum_{k=\ell+1}^{n}\mathrm{Cov}\left(Y_{j}, Y_{k}\right)$$ for $n\rightarrow\infty$ and $\ell$ fixed?