Let $X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ be i.i.d. Gaussian random variables of mean zero and variance one.
Then $$Z:= \sum_i \frac{X_i Y_i}{\sqrt{n}}$$ is approximately Gaussian with $$\mathbf E Z =0, \quad \mathbf E Z^2=1.$$ The central limit theorem implies that $$\mathbf E f(Z) =\frac{1}{\sqrt{2\pi}} \int_{\mathbf R} f(x) e^{-x^2/2}\,\mathrm{d}x + o(1)$$ for any smooth bounded $f$ as $n\to\infty$. Due to the Berry-Essen theorem the convergence rate should be $n^{-1/2}$.
Question: Can the subleading term be identified? I.e. can one get $$\mathbf E f(Z) =\frac{1}{\sqrt{2\pi}} \int_{\mathbf R} f(x)e^{-x^2/2}\,\mathrm{d}x + \frac{F(f)}{\sqrt{n}} + o(n^{-1/2})$$ for some $F(f)$? If yes, does this $F(f)$ depend on the derivative of $f$?