On an equation with prime numbers close to squares

Let $[\cdot ]$ be the floor function. In this paper, we show that when $1<c<37\mathrm{/}36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3],, \end{equation*} where $p_1,p_2,p_3$ are primes close to squares.