  We estimate mixed character sums of polynomial values over elements of a
finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered
basis over the subfield $\mathbb F_q$. First we use a combination of the
inclusion-exclusion principle with bounds on character sums over linear
subspaces to get nontrivial bounds for large $q$. Then we focus on the
particular case $q=2$, which is more intricate. The bounds depend on certain
natural restrictions. We also provide families of examples for which the
conditions of our bounds are fulfilled. In particular, we completely classify
all monomials as argument of the additive character for which our bound is
applicable. Moreover, we also show that it is applicable for a large family of
rational functions, which includes all reciprocal monomials.
