For sufficiently large n with r = floor((n-1)/2), any Hamiltonian Berge cycle plus one additional edge in an n-vertex r-uniform hypergraph contains Berge cycles of all lengths 2 to n.
Green,A Szemerédi-type regularity lemma in abelian groups, with applications, Geom
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One extra edge forces Berge pancyclicity
For sufficiently large n with r = floor((n-1)/2), any Hamiltonian Berge cycle plus one additional edge in an n-vertex r-uniform hypergraph contains Berge cycles of all lengths 2 to n.