On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$

For an integer $k\geq 2$, let $\{F^{(k)}_{n} \}_{n\geq 0}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ having at least two representations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$.