A hybrid of two theorems of Piatetski-Shapiro

Let $c > 1$ and $0 < \gamma < 1$ be real, with $c \notin \mathbb N$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2, \dots, p_s$ of index $\gamma$---that is, primes of the form $p = \lfloor m^{1/\gamma} \rfloor$ for some $m \in \mathbb N$.