Flexible Team Orienteering Problem
I don't know if it has a name but it's NP-hard. Just take $K=1$, give every edge and vertex weight&nbsp;$1$ and ask if there's a solution with vertex reward&nbsp;$|V|$ and edge cost at most $|V|-1$. This happens if, and only if, there's an $s$&ndash;$t$ Hamiltonian path.  The decision version "Can I get reward at least&nbsp;$R$ with cost at most&nbsp;$C$" is NP-complete.