Subset sums of power values over Galois rings

Let $R=GR(p^m,p^{mf})$, where $p$ is prime, $q=p^f$, $m,f\ge 1$, $d\ge 2$ and $p\nmid d$. We count $n$-element subsets of $P=\{x^d:x\in R\}\setminus \{0\}$, the set of distinct nonzero $d$th-power values, with a prescribed sum. For fixed $m,d,n$ with $n\ge 3$, the count equals $$q^{-m}\left(\genfrac{}{}{0px}{}{\mathrm{\mid }P\mathrm{\mid }}{n}\right)(1+O_{m,d,n}(q^{1-n\mathrm{/}2}))$$ as $q\to \mathrm{\infty }$ through admissible prime powers, uniformly over all targets and with both $p$ and $f$ allowed to vary. For fixed $m,d$, we also classify uniform relative equidistribution at the remaining positive sizes: it fails at $n=1$, and at $n=2$ it holds along an admissible sequence if and only if, for all sufficiently large terms, $\gcd (d,q-1)=1$ and $(m\ge 2\text{ or }p\mathrm{\ne }2)$.