On a Diophantine inequality involving primes yielding square-free sums with given integers

Let \alpha\in \mathbb{R}\setminus\mathbb{Q} and \beta\in \mathbb{R} be given. Suppose that a_{1},\ldots,a_{s} are distinct positive integers that do not contain a reduced residue system modulo p^{2} for any prime p . We prove that there exist infinitely many primes p such that the inequality \|\alpha p+\beta\|<p^{-1/10} holds and all the numbers p+a_{1},\ldots,p+a_{s} are square-free.