  We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the
form $(\tau x+c) \omega(n)+xn$, $n=1, 2, \ldots$, with fixed $\omega(T) \in
\mathbb{Z}[T]$ and $\tau \in \mathbb{R}$, which holds for almost all $c\in
[0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of
M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this
question and some classical partial differential equations. We extend this to
more general settings of families of polynomials $xn+y \omega(n)$ for all
$(x,y)\in [0,1)^2$ with $f(x,y)=z$ for a set of $z \in [0,1)$ of full Lebesgue
measure, provided that $f$ is some H\"older function.
