I was working through a paper on efficient congruencing when I came across the following, but I can't understand why it is correct. For reference $\xi$ is an integer and $1 \leq \xi \leq p$ where $p$ is a prime. My instinct is that it is Hensel's Lemma related, but I haven't been able to find any resources to help me with that.
Assuming $\textbf{x}$ to be well-conditioned, that is, $x_1, \dots, x_k$ are in distinct congruence classes modulo $p$ then for a fixed $\textbf{n} \in \mathbb{Z}^k$ we can lift solutions to \begin{equation*} \sum_{i = 1}^k (x_i - \xi)^j \equiv n_j \quad (mod\ p) \quad (1 \leq j \leq k) \end{equation*} With $1 \leq \textbf{x} \leq p$ uniquely to solutions to \begin{equation*} \sum_{i = 1}^k (x_i - \xi)^j \equiv n_j \quad (mod\ p^k) \quad (1 \leq j \leq k) \end{equation*} With $1 \leq \textbf{x} \leq p^k$.