On powers of tight Hamilton cycles in randomly perturbed hypergraphs

For integers (Figure presented.) and (Figure presented.), we show that for every (Figure presented.), there exists (Figure presented.) such that the union of (Figure presented.) -uniform hypergraph on (Figure presented.) vertices with minimum codegree at least (Figure presented.) and a binomial random (Figure presented.) -uniform hypergraph (Figure presented.) with (Figure presented.) on the same vertex set contains the (Figure presented.) power of a tight Hamilton cycle with high probability. Moreover, a construction shows that one cannot take (Figure presented.), where (Figure presented.) is a constant. Thus the bound on (Figure presented.) is optimal up to the value of (Figure presented.) and this answers a question of Bedenknecht, Han, Kohayakawa, and Mota.