On small fractional parts of perturbed polynomials

Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on earlier work by Madritsch and Tichy. In particular, let $f=P+\phi$ where $P$ is a polynomial of degree $k$ and $\phi$ is a linear combination of functions of shape $x^c$, $c\not \in \mathbb{N}$, $1<c<k$. We prove that for any given irrational $\xi$ we have $$\min_{\substack{2\leq p\leq X\\ p \text{ prime}}} \Vert \xi \lfloor f(p)\rfloor\Vert \ll_{f,\epsilon} X^{-\rho(k)+\epsilon},$$ for $P$ belonging to a certain class of polynomials and with $\rho(k)>0$ being an explicitly given rational function in $k$.