A binary additive equation with prime and square-free number

Let $[\cdot ]$ be the floor function. In this paper, we show that when $1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c]+[m^c],, \end{equation*} where $p$ is a prime and $m$ is a square-free.