On the problem of Pillai with $k$--generalized Fibonacci numbers and powers of $3$

For an integer $k\ge 2$, let $\{F_n^{(k)}{\}}_{n\ge 2-k}$ be the $k$–generalized Fibonacci sequence which starts with $0,\dots ,0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ with at least two representations as a difference between a $k$-generalized Fibonacci number and a power of $3$. This paper continues the previous work of the first author for the Fibonacci numbers, and the Tribonacci numbers.