  Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors,
counted according to multiplicity. In this paper, it is proved that for
$\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and
$0<\theta<10/1561$, there exist infinitely many primes $p$, such that $\|\alpha
p^2+\beta\|<p^{-\theta}$ and $p+2=\mathcal{P}_4$, which constitutes an
improvement upon the previous result.
