  We consider variants of the classic Multiway Cut problem. Multiway Cut asks
to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals.
Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway,
a generalization of the problem, in which the goal is to minimize the $\ell_p$
norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2}
n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon
the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to
Chandrasekaran and Wang.
  We also introduce and study Norm Multiway Cut, a further generalization of
Multiway Cut. We assume that we are given access to an oracle, which answers
certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$
approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2}
k)$ approximation algorithm with a stronger oracle. Additionally, we show that
without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation
algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random
Conjecture.
