  Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge
belongs to $R$. A hypergraph $\mathcal{H}$ is called \textit{covering} if every
vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we
show that every covering $[3]$-uniform hypergraph on $n\geq 6$ vertices
contains a Berge cycle $C_s$ for any $3\leq s\leq n$. As an application, we
determine the maximum Lagrangian of $k$-uniform Berge-$C_{t}$-free hypergraphs
and Berge-$P_{t}$-free hypergraphs.
