  Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic
field of odd residual characteristic. The goal of this paper is to describe the
supercuspidal representations within certain L-packets of $\mathbf{G}$, which
are classified by Arthur and Mok using the theory of endoscopy. The description
is given in terms of the cuspidal types constructed by Bushnell-Kutzko and
Stevens. As a starting example, we require the parameters of our packets to
satisfy certain regularity conditions, such that these packets consist of very
cuspidal representations in the sense of Adler and Reeder. To achieve our goal,
we first interpret the question as to study the reducibilities of some
parabolically induced representations, using a theory of M{\oe}glin and
Shahidi; we then apply a relation, given by Blondel, between these
reducibilities and the structures of some Hecke algebras, where the latter can
be computed using a Theorem of Lusztig. We can interpret our final result as
explicitly describing the local Langlands correspondence for $\mathbf{G}$.
