Diophantine approximation with mixed powers of Piatetski-Shapiro primes

Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $\eta$ is real and the constants $\lambda _i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<\gamma<1$ and $\theta>0$, there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda _1p_1 + \lambda _2p_2 + \lambda _3p^2_3+\eta|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64\gamma}{52}}+\theta} \end{equation*} and such that $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3$.