  For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for
any abelian group $G$. The best known bounds relating the parameters in the
lemma for general $G$ are of tower-type. For $k>3$, even in the case
$G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. This
special case has received considerable attention due to its close connection to
property testing of boolean functions. For every $k\geq 3$, we prove a
polynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ is
any fixed prime. This extends the result for $k=3$ by the first two authors.
Due to substantial issues with generalizing the proof of the $k=3$ case, a new
strategy is developed in order to prove the result for $k>3$.
