Let $\mathbb{F}$ be an finite field. Fix some enumeration of it, i.e. $\mathbb{F}=\{f_1,\dots,f_{|\mathbb{F}|}\}$, so that we can talk about the $i^{\text{th}}$ element in the field.
Definition: A degree $k$ curve over $\mathbb{F}^m$ is a mutlti-set $$\{u_0+tu_1+\cdots+t^ku_k:t\in\mathbb{F}\}$$ where $u_0,\dots,u_k\in\mathbb{F}^m$.
Clearly, $\{u_0+tu_1+\cdots+t^ku_k:t\in\mathbb{F}\}=\{u_0+ctu_1+\cdots+c^kt^ku_k:t\in\mathbb{F}\}$ for every $c\in\mathbb{F}\setminus\{0\}$. So, we fix some canonical rep. for each curve.
Thanks to the enumeration defined on $\mathbb{F}$ we can talk about "the $i^{\text{th}}$ point" in a curve.
Defintion: Let $x_1,\dots,x_k\in\mathbb{F}^m$ (not necessarily distinct). The set $P(\langle x_1,\dots,x_k\rangle)$ is the set of all degree $k$ curves over $\mathbb{F}^m$ whose first $k$ points are $x_1,\dots,x_k$.
Problem: Let $S\subseteq \mathbb{F}^m$ be any set and let $x_1,\dots,x_k\in\mathbb{F}^m$ (not all the same). Prove that the average of $\frac{|C\cap S|}{|C|}$ taken over all $C\in P(\langle x_1,\dots,x_k\rangle)$, is $\frac{|S|}{|\mathbb{F}|^m}$, i.e. $$\mathbb{E}\left[\frac{|C\cap S|}{|C|}\right]=\frac{|S|}{|\mathbb{F}|^m}$$ where $C$ is uniformly drawn from $P(\langle x_1,\dots,x_k\rangle)$.
Initial Direction: Someone told me to look at the following enumeration of $P(\langle x_1,\dots,x_k\rangle)$:
For every $x\in\mathbb{F}^m$ and every $k<j\leq |\mathbb{F}|$ the curve in $P(\langle x_1,\dots,x_k\rangle)$ whose $j^{\text{th}}$ point is $x$.
This counts each curve in $P(\langle x_1,\dots,x_k\rangle)$ and each point in $\mathbb{F}^m$ exactly $|\mathbb{F}|-k$ times. But, I don't know how to proceed.
Edit: The lemma comes from a paper, in page 156 of the pdf, the relevant claim is A.9, and it is primarily used in page 67 of the pdf.