jats:titleAbstract</jats:title>jats:pFor jats:inline-formulajats:alternativesjats:tex-math$$0\le \ell <k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mn0</mml:mn> mml:mo≤</mml:mo> mml:miℓ</mml:mi> mml:mo<</mml:mo> mml:mik</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, a Hamilton jats:inline-formulajats:alternativesjats:tex-math$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-cycle in a jats:italick</jats:italic>-uniform hypergraph jats:italicH</jats:italic> is a cyclic ordering of the vertices of jats:italicH</jats:italic> in which the edges are segments of length jats:italick</jats:italic> and every two consecutive edges overlap in exactly jats:inline-formulajats:alternativesjats:tex-math$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> vertices. We show that for all jats:inline-formulajats:alternativesjats:tex-math$$0\le \ell <k-1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mn0</mml:mn> mml:mo≤</mml:mo> mml:miℓ</mml:mi> mml:mo<</mml:mo> mml:mik</mml:mi> mml:mo-</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, every jats:italick</jats:italic>-graph with minimum co-degree jats:inline-formulajats:alternativesjats:tex-math$$\delta n$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miδ</mml:mi> mml:min</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with jats:inline-formulajats:alternativesjats:tex-math$$\delta >1/2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miδ</mml:mi> mml:mo></mml:mo> mml:mn1</mml:mn> mml:mo/</mml:mo> mml:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> has (asymptotically and up to a subexponential factor) at least as many Hamilton jats:inline-formulajats:alternativesjats:tex-math$$\ell $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miℓ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-cycles as a typical random jats:italick</jats:italic>-graph with edge-probability jats:inline-formulajats:alternativesjats:tex-math$$\delta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miδ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>. This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values jats:inline-formulajats:alternativesjats:tex-math$$0\le \ell <k-1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mn0</mml:mn> mml:mo≤</mml:mo> mml:miℓ</mml:mi> mml:mo<</mml:mo> mml:mik</mml:mi> mml:mo-</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>