  In the Max $k$-Weight SAT (aka Max SAT with Cardinality Constraint) problem,
we are given a CNF formula with $n$ variables and $m$ clauses together with a
positive integer $k$. The goal is to find an assignment where at most $k$
variables are set to one that satisfies as many constraints as possible.
Recently, Jain et al. [SODA'23] gave an FPT approximation scheme (FPT-AS) with
running time $2^{O\left(\left(dk/\epsilon\right)^d\right)} \cdot (n +
m)^{O(1)}$ for Max $k$-Weight SAT when the incidence graph is $K_{d,d}$-free.
They asked whether a polynomial-size approximate kernel exists. In this work,
we answer this question positively by giving an $(1 - \epsilon)$-approximate
kernel with $\left(\frac{d k}{\epsilon}\right)^{O(d)}$ variables. This also
implies an improved FPT-AS with running time $(dk/\epsilon)^{O(dk)} \cdot (n +
m)^{O(1)}$. Our approximate kernel is based mainly on a couple of greedy
strategies together with a sunflower lemma-style reduction rule.
