On the representation of integer as sum of a square-free number and a prime of special type

We prove that there are infinitely many integers, which can represent as sum of a square-free integer and a prime $p$ with $\mathrm{\mid }\mathrm{\mid }\alpha p+\beta \mathrm{\mid }\mathrm{\mid }<p^{-1\mathrm{/}10}$, where $\alpha$ is irrational.