In this case:

$O(k)=O(klogk)$

The query complexity used is $O(1)$ and $O(poly(logn))$.

The query complexity used in this paper is $O(1)$ and $O(poly(logn))$.

For Lemma 3.1 there is a note that the query complexity used is $O(1)$.

If the question is how Lemma 3.1 generalizes to non-constant query complexity, this does present a problem outside of $O(poly(f(n)))$.

This problem is sidestepped by composing a verifier that reduces query complexity to $O(1)$ (Lemma 4.4).