= Solution
The <finite prime-power Teichmuller representative> has an explicit description. Let $y$ be the unique least nonnegative residue satisfying
$$
\boxed{y\equiv x^{p^{n-1}}\pmod{p^n}.}
$$
Repeated <Fermat's little theorem> gives $y\equiv x\pmod p$. Part (ii) gives
$$
y^p\equiv x^{p^n}\equiv x^{p^{n-1}}\equiv y\pmod{p^n},
$$
so this residue has both required properties.
For uniqueness, suppose $z$ also has them. From $z^p\equiv z\pmod{p^n}$, raising congruences to the $p$th power repeatedly gives $z^{p^j}\equiv z\pmod{p^n}$ for every $j\geq0$. Meanwhile $z\equiv x\pmod p$ and the lifting result with $r=n-1$ imply
$$
z^{p^{n-1}}\equiv x^{p^{n-1}}\pmod{p^n}.
$$
Hence $z\equiv y\pmod{p^n}$. Both lie between zero and $p^n-1$, so $z=y$. Thus \b[existence and uniqueness hold for every residue class, including zero], without needing a general <Hensel lemma>.
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