  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 $k=3$.
