= Solution
Every constructible <set> is well-orderable: the <constructible hierarchy> has a canonical well-order, obtained by ordering the stage of first appearance, <logical formula> codes and the already ordered finite parameter lists. Its restriction to any <set> belonging to $L$ is a well-order in $L$ and remains a well-order in the ambient universe. This does not imply that arbitrary ambient <sets> are well-orderable.
If $a\in L$ is finite, all its <subsets> are finite combinations of its elements and hence constructible. Moreover the <finite set> of all those <subsets> is constructible by the <finite set> operations available in $L$. Thus the full ambient <power set> $\mathcal P(a)$ belongs to $L$.
Conversely, suppose $a\in L$ is infinite and that its full ambient <power set> belonged to $L$. A well-order of $a$ supplies an <injection> $j:\omega\to a$ by taking its first countably many elements. The map
$$
\mathcal P(\omega)\to\mathcal P(a),\qquad u\longmapsto j[u]
$$
is <injective>. But $\mathcal P(a)\in L$ would make that <set> well-orderable in the ambient universe, and pulling its well-order back along this <injection> would well-order $\mathcal P(\omega)$. This contradicts the hypothesis. Hence
$$
\boxed{a\in L:\qquad\mathcal P^{V}(a)\in L\quad\Longleftrightarrow\quad a\text{ is finite}.}
$$
The superscript emphasizes that the source's final claim concerns the ambient <power set>. The internal <power set> $\mathcal P^L(a)$ exists in $L$ for every $a\in L$ by part (ii), including infinite ones.
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