For each k≥ 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= 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≥ 3, we prove a polynomial bound relating the parameters for G= 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.