= Solution
Binomial <factorial moments> give $\langle N_1(N_1-1)\rangle=N(N-1)p^2$, its mode-zero counterpart $N(N-1)(1-p)^2$, and $\langle N_0N_1\rangle=N(N-1)p(1-p)$. Consequently the exact projected expectation is
$$
\frac{E(\chi)}N=\mathcal E p+\frac{U(N-1)}{4\pi}[1+2p(1-p)].
$$
Since $p(1-p)=\sin^2\chi/4$ and $n=N/(2\pi)$,
$$
\boxed{\frac{E(\chi)}N\simeq\mathcal E\sin^2(\chi/2)+\frac{nU}{2}\left[1+\frac12\sin^2\chi\right]}.
$$
The finite-$N$ correction is simply the replacement $N\mapsto N-1$ in the interaction term. Number correlations must be retained to obtain that correction.
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