  We give a reduction formula for the Waring number $g(k,q)$ over a finite
field $\mathbb{F}_q$. By exploiting the relation between $g(k,q)$ with the
diameter of the generalized Paley graph $\Gamma(k,q)$ and by using the
characterization due to Pearce and Praeger (2019) of those $\Gamma(k,q)$ which
are Cartesian decomposable, we obtain the reduction formula
  $$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$ for $p$ prime
and $a,b,c$ positive integers under certain arithmetic conditions. Then, we
find some arithmetic conditions to apply the formula above, which allow us to
obtain many infinite families of explicit values of Waring numbers. Finally, we
use the reduction formula together with the characterization of $2$-weight
irreducible cyclic codes due to Schmidt and White (2002) to find infinite
families of explicit even values of $g(k,q)$.

Number Theory;Combinatorics