Further bounds in the polynomial Szemer\'{e}di theorem over finite fields

We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$x,x+y,\mathrm{.}\mathrm{.}\mathrm{.},x+(m-1)y,x+y^m,\mathrm{.}\mathrm{.}\mathrm{.},x+y^{m+k-1}\mathrm{.}$$ 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 $$x,x+y^k,\mathrm{.}\mathrm{.}\mathrm{.},x+(m-1)y^k\mathrm{.}$$ Both results follow from an estimate of the number of such progressions in an arbitrary subset of a finite field.