The following is not the answer of your question. But I think it is what you really want.

I guess you might be figuring out Leo's proof of McLaughlin's conjecture and his answer to Question 65 in Harvey's problem collection paper. The point is that by applying a nonstandard ordinal, Leo obtained a nonstandard $\Pi^0_1$-singleton so that it is not hyperarithmetic. 



There are several ways to see above. One is by applying Barwise compactness. Another is to use Gandy's basis. By either way, you may obtain a nonstandard $\omega$-model $M$ of KP with $\omega_1^M=\omega_1^{CK}$ in which there is a $\Pi^0_1$-singleton $x$ which is not  hyperarithmetic (in the real universe sense).

(To see it by Gandy's basis theorem. Just apply it to obtain an $\omega$-model $M\models KP$ in which $\omega_1^{CK}$ is nonstandard. In $M$, by fixing a nonstandard recursive ordinal $\alpha$, we perform Leo's proof that there is $\Pi^0_1$-singleton $x$ not recursive in $\emptyset^{\alpha}$.)

Now take $N=L_{\omega_1^{CK}}[x]$. Since $x\leq_h M$, we have that $N\models KP$. It is not difficult to see that $x$ is also a $\Pi^0_1$-singleton in $N$. However, $N\models$`` $x$ is not hyperarithmetic" since $N$ is a well-founded model.