The problem
Let $\mathbb F$ be a finite field and $d < | \mathbb F |$. (Can think of the setting $d \approx |\mathbb F|/2$)
Let $$ A = \left\{ \frac{p(x)}{q(x)} \ \colon \ p,q \in \mathbb F[x] \text{ and } \mathrm{deg}(q), \mathrm{deg}(p) \leq d \right\} \setminus \left\{p(x) \in \mathbb F \colon \mathrm{deg}(p) \leq d\right\}$$ That is, $A$ is the set of all univariate rational functions with both numerator and denominator polynomials of degree at most $d$, who are not low-degree themselves. Note that the degree of $q$ need not be smaller than that of $p$; for example, $1/x$ is an interesting function to us.
What is the cardinality of $|A|$?
A definitional issue
Since I'm only interested in counting, I'll choose a special symbol $* \notin \mathbb F$ and let $\frac{p(x)}{q(x)} = *$ whenever $q(x) = 0$. If there is a more elegant or correct suggestion for dealing with $q(x)=0$ I'm happy to consider it.