  We study the $p$-adic absolute value of the roots of the $L$-functions
associated to certain twisted character sums, and additive character sums
associated to polynomials $P(x^d)$, when $P$ varies among the space of
polynomial of fixed degree $e$ over a finite field of characteristic $p$. For
sufficiently large $p$, we determine in both cases generic Newton polygons for
these $L$-functions, which is a lower bound for the Newton polygons, and the
set of polynomials of degree $e$ for which this generic polygon is attained. In
the case of twisted sums, we show that the lower polygon defined in \cite{as1}
is tight when $p\equiv 1 [de]$, and that it is the actual Newton polygon for
any degree $e$ polynomial.
