On small fractional parts of polynomial-like functions

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for $f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and any fixed $\xi \in [0,1]$ they obtained $$\underset{2\le p\le X}{\min }\mathrm{\parallel }\xi \lfloor f(p)\rfloor \mathrm{\parallel }{\ll }_{k,c,ϵ}{X}^{-{\rho }_1(c,k)+ϵ}$$ for ${\rho }_1(c,k)>0$ explicitly given. In the present note, we improve upon their results in the case $c>k$ and $c>4$.