The system $\textbf{ACA}_0' $ is the theory $ \textbf{ACA}_0 $ plus the statement $\forall n\,\forall X \,\exists X^{(n)}$. I'm not entirely sure what the proper way to express this as a second order arithmetic formula. The reason for this is that to define the $n$-th jump we need to make use of sequences of set so that we can recursively define the $i$-th jump for $i\leq n$. I thought that it could be expressed as: \begin{equation*} \forall n\,\forall A\, \exists X_0\oplus\dots \oplus X_n\,(X_0=A\wedge \forall i< n X_{i+1}=X_i') \end{equation*} (This formula still makes use of many abbreviations and is not a proper second order formula but I would know how to translate it into a true second order formula). So the existence of something like the Turing join of the first $n$ jumps of a set $X$ is trivial for the system of $\textbf{ACA}_0'$.
For the Hirschfeldt exercise it asks to show that $\textbf{ACA}_0' $ proves for every arithmetic formula $\varphi(x,Y)$ where $Y$ is the only set variable that appears and is unbounded and for every set $X$ and number $n$ we have that there is a sequence of sets $X_0,\dots X_n$ such that $X_0=X$ and for all $i<n$ $X_{i+1}=\{x:\varphi(x,X_i)\}$. I am assuming that by sequence of sets Hirschfeldt is refering to a set of the form $X_0\oplus\dots\oplus X_n$.
For the actual solution of the exercise my attempt doesn't seem very formal. The formula $\varphi(x,Y)$ will be of complexity $\Delta^0_{k+1}(Y)$ meaning that the set $\{x:\varphi(x,Y)\}\leq_T Y^{(k)}$ by Post's Theorem. The sequence I want should therefore be computable from $X^{(k\cdot n)}$ in fact we have that the sequence needed for the exercise $X_0,X_1,\dots X_n$ is such that for all $i\leq n$ we have $X_i\leq_T X^{(i\cdot k)}$ and so I should also have $X_0\oplus\dots\oplus X_n\leq_T X^{(0)}\oplus X^{(k)}\oplus\dots\oplus X^{(n\cdot k)}$. This argument feels very hand wavy and I would like for it to be a bit more rigorous.
Edit: after some thought I think I can state a bit clearly my question. Given $\varphi(n,Y)$ an arithmetic formula of complexity $\Delta^0_{k+1}$ and define $Y^+=\{n:\varphi(n,Y)\}$. Is there an effective way to find a sequence of indices $e_0,e_1,\dots$ such that $Y^+=\Phi^{Y^{(k)}}_{e_0}$ and in general we have that $(Y^+)^{(i\cdot k)}=\Phi^{Y^{((i+1)\cdot k)}}_{e_i}$ ? In this case I believe the problem is done because I can express the sequence $X_0\oplus\dots X_n$ arithmetically relative to $X^{(k)}\oplus\dots\oplus X^{(k\cdot n)}$.
Edit: I think I can reduce the problem to something more simple. There exists an index $e_1$ such that for all $A'$ $\Phi^{A'}_{e_1}$ is the characteristic function of $A$. By induction I can show in $\textbf{ACA}_0'$ that for each $k$ there is an index $e_k$ such that for all $A$ I can calculate uniformly $A$ from the $k$-th jump. This is just a consequence of the smn theorem.