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We prove that, for any   $c_{1},c_{2},c_{3}\in (1,41/35)$  , every sufficiently large odd number   $N$   can be represented as the sum of three primes,   $N = p_{1} + p_{2} +p_{3}$  , such that   $p_{i} = \lfloor n_{i}^{c_{i}}\rfloor $   for some   $n_{i} \in \mathbb N$   for each   $1 \leq i \leq 3$  . Our arguments are based on a variant of Green’s transference principle due to Matomäki, Maynard, and Shao. We prove a necessary restriction estimate using Bourgain’s strategy and employ Harman’s sieve method to optimize our upper bound for   $c_{i}$  .

Vinogradov’s Theorem with Piatetski-Shapiro Primes