Let k,a,b be positive integers with a+b=k. A k-uniform hypergraph is called an (a,b)-cycle if there is a partition (A0,B0,A1,B1,…,At−1,Bt−1) of the vertex set with |Ai|=a, |Bi|=b such that Ai∪Bi and Bi∪Ai+1 (subscripts module t) are edges for all i=0,1,…,t−1. Let H be a k-uniform n-vertex hypergraph with n≥5k and n divisible by k. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists α∈(0,1) such that δa(H)≥(α+o(1))(n−ab) and δb(H)≥(1−α+o(1))(n−ba), then H contains a Hamilton (a,b)-cycle. As a corollary, we prove that if δℓ(H)≥(1/2+o(1))(n−ℓk−ℓ) for some ℓ≥k/2, then H contains a Hamilton (k−ℓ,ℓ)-cycle and this is asymptotically best possible.