What's the $p$-th power of a $k$-uniform tight cycle?

Question

I understand that in a graph when there's a cycle, the $p$-th power of the cycle is when all vertices at a distance of at most $p$ from each other (on the base cycle) are joined with edges. A $k$-uniform tight cycle is a cycle where all edges have $k$ vertices and any two consecutive edges overlap in $k-1$ vertices. Could someone explain how the $p$-th power of a $k$-uniform tight cycle is defined? I found a definition that says it is the $k$-graph obtained by replacing every edge of the $(k+p-1)$-uniform tight cycle by $K_{k+p-1}^{(k)}$, but I'm not sure how this relates to the definition I know for graphs.