Let $ [x] $ denote the integer part of $ x\in\mathbb{R} $ , and let $ \|x\| $ denote the distance from $ x $ to the nearest integer.In this paper, we prove that if $ \alpha $ is irrational and $ \beta $ is an arbitrary real, then for every fixed $ \frac{47}{48}<\gamma<1 $ , there exist infinitely many primes $ p $ of the form $ p=[n^{1/\gamma}] $ satisfying the inequality $ \|\alpha p^{3}+\beta\|<p^{\frac{47-48\gamma}{96}+\varepsilon} $ .
