  Let $\alpha,\beta$ be real numbers such that $\alpha>1$ is irrational and of
finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In
this paper, it is shown that there are infinitely many Piatetski-Shapiro primes
$p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence
$\big(\left\lfloor\alpha m+\beta\right\rfloor\big)_{m=1}^\infty$.
