On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

Let $\{F_n{\}}_{n\ge 0}$ be the sequence of Fibonacci numbers and let $p$ be a prime. For an integer $c$ we write $m_{F,p}(c)$ for the number of distinct representations of $c$ as $F_k-p^{\mathrm{\ell }}$ with $k\ge 2$ and $\mathrm{\ell }\ge 0$. We prove that $m_{F,p}(c)\le 4$.