  We provide upper bounds for the size of subsets of finite fields lacking the
polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$
These are the first known upper bounds in the polynomial Szemer\'{e}di theorem
for the case when polynomials are neither linearly independent nor homogeneous
of the same degree. We moreover improve known bounds for subsets of finite
fields lacking arithmetic progressions with a difference coming from the set of
$k$-th power residues, i.e. configurations of the form $$ x, x+y^k, ...,
x+(m-1)y^k.$$ Both results follow from an estimate of the number of such
progressions in an arbitrary subset of a finite field.
