  Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$
with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we
obtain new results concerning the number of solutions to congruences of the
form $$y\equiv f(x) \pmod{q},$$ with integer variables lying in some cube
$\mathcal B$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev,
Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value
theorem and a lattice point counting problem. We treat the lattice point
problem differently using transference principles from the Geometry of Numbers.
We also use a variant of the main conjecture for the Vinogradov mean value
theorem of Bourgain, Demeter and Guth and of Wooley which allows one to deal
with rather sparse sets.
