Counting subspaces with prescribed Welch sums

Let $q=2^n$ with $n=2m+1$ for an integer $m\ge 1$, and let ${\mathbb{F}}_q$ be the finite field with $q$ elements. Let $W_n(x)=x^{2^m+3}$ be the Welch function on ${\mathbb{F}}_q$. For $1\le k\le n$ and $b\in {\mathbb{F}}_q$, denote by $N_{n,k}(b)$ the number of $k$-dimensional ${\mathbb{F}}_2$-subspaces $E$ of \mathbb F}_q such that $$\sum _{x\in E}{W}_n(x)=b\mathrm{.}$$ In this paper, we determine $N_{n,k}(b)$ for all $k$. The only nontrivial case is $k=3$, for which the answer is given in terms of a binary Kloosterman sum. As a consequence, we prove a recent conjecture of Hou and Zhao on the third-order sum-freedom of the Welch function.