Normality of the Thue-Morse function for finite fields along polynomial values

Let ${\mathbb F}_q$ be the finite field of $q$ elements, where $q=p^r$ is a power of the prime $p$, and $\left(\beta_1, \beta_2, \dots, \beta_r \right)$ be an ordered basis of ${\mathbb F}_q$ over ${\mathbb F}_p$. For $$\xi=\sum_{i=1}^rx_i\beta_i, \quad \quad x_i\in{\mathbb F}_p,$$ we define the Thue-Morse or sum-of-digits function $T(\xi)$ on ${\mathbb F}_q$ by $$T(\xi)=\sum_{i=1}^{r}x_i.$$ For a given pattern length $s$ with $1\le s\le q$, a subset ${\cal A}=\{\alpha_1,\ldots,\alpha_s\}\subset {\mathbb F}_q$, a polynomial $f(X)\in{\mathbb F}_q[X]$ of degree $d$ and a vector $\underline{c}=(c_1,\ldots,c_s)\in{\mathbb F}_p^s$ we put $${\cal T}(\underline{c},{\cal A},f)=\{\xi\in{\mathbb F}_q : T(f(\xi+\alpha_i))=c_i,~i=1,\ldots,s\}.$$ In this paper we will see that under some natural conditions, the size of~${\cal T}(\underline{c},{\cal A},f)$ is asymptotically the same for all~$\underline{c}$ and ${\cal A}$ in both cases, $p\rightarrow \infty$ and $r\rightarrow \infty$, respectively. More precisely, we have $$\left||{\cal T}(\underline{c},{\cal A},f)|-p^{r-s}\right|\le (d-1)q^{1/2}$$ under certain conditions on $d,q$ and $s$. For monomials of large degree we improve this bound as well as we find conditions on $d,q$ and $s$ for which this bound is not true. In particular, if $1\le d<p$ we have the dichotomy that the bound is valid if $s\le d$ and fails for some $\underline{c}$ and ${\cal A}$ if $s\ge d+1$. The case $s=1$ was studied before by Dartyge and S\'ark\"ozy.