  The $\boldsymbol{v}$-function is a key ingredient in the wild McKay
correspondence. In this paper, we give a formula to compute it in terms of
valuations of Witt vectors, when the given group is a cyclic group of prime
power order. We apply it to study singularities of a quotient variety by a
cyclic group of prime square order. We give a criterion whether the stringy
motive of the quotient variety converges or not. Furthermore, if the given
representation is indecomposable, then we also give a simple criterion for the
quotient variety being terminal, canonical, log canonical, and not log
canonical.
