  Building upon the observation that the newly defined~\cite{EFRST20} concept
of $c$-differential uniformity is not invariant under EA or
CCZ-equivalence~\cite{SPRS20}, we showed in~\cite{SG20} that adding some
appropriate linearized monomials increases the $c$-differential uniformity of
the inverse function, significantly, for some~$c$. We continue that
investigation here. First, by analyzing the involved equations, we find bounds
for the uniformity of the Gold function perturbed by a single monomial,
exhibiting the discrepancy we previously observed on the inverse function.
Secondly, to treat the general case of perturbations via any linearized
polynomial, we use characters in the finite field to express all entries in the
$c$-Differential Distribution Table (DDT) of an $(n,n)$-function on the finite
field $\F_{p^n}$, and further, we use that method to find explicit expressions
for all entries of the $c$-DDT of the perturbed Gold function (via an arbitrary
linearized polynomial).
