  We show that if there exists an integer subject to some congruence conditions
that cannot be written as the sum of the norm of an ideal in
$\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then
there are infinitely many such integers. Also, if there exists an integer that
cannot be written as the sum of an integer which is the norm of an ideal in in
$\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$
is a prime, then there are infinitely many such integers. Finally it is shown
that there are infinitely many integers not the sum of the norm of an ideal in
$\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, for $p\geq 3$
prime.
