  We give a new construction of $p$-typical Witt vectors with coefficients in
terms of ghost maps and show that this construction is isomorphic to the one
defined in terms of formal power series from the authors' previous paper. We
show that our construction recovers Kaledin's polynomial Witt vectors in the
case of vector spaces over a perfect field of characteristic $p$. We then
identify the components of the $p$-typical TR with coefficients, originally
defined by Lindenstrauss and McCarthy and later reworked by the second and
third authors in joint work with McCandless, with the $p$-typical Witt vectors
with coefficients. This extends a celebrated result of Hesselholt and
Hesselholt-Madsen relating the components of TR with the Witt vectors. As an
application, we given an algebraic description of the components of the
Hill-Hopkins-Ravenel norm for cyclic $p$-groups in terms of $p$-typical Witt
vectors with coefficients.
