The k–generalized Pell sequence P(k):=(Pn(k))n≥-(k-2)\documentclass[12pt]{minimal}
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\begin{document}$$P^{(k)}:=(P_n^{(k)})_{n\ge -(k-2)}$$\end{document} is the linear recurrence sequence of order k, whose first k terms are 0,…,0,1\documentclass[12pt]{minimal}
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\begin{document}$$0, \ldots , 0,1$$\end{document} and satisfies the relation Pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k)\documentclass[12pt]{minimal}
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\begin{document}$$ P_{n}^{(k)} =2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}$$\end{document}, for all n,k≥2\documentclass[12pt]{minimal}
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\begin{document}$$ n,k\ge 2 $$\end{document}. In this paper, we investigate about integers that have at least two representations as a difference between a k–Pell number and a perfect power. In order to exhibit a solution method when b is known, we find all the integers c that have at least two representations of the form Pn(k)-bm\documentclass[12pt]{minimal}
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\begin{document}$$ P_{n}^{(k)} - b^{m}$$\end{document} for b∈[2,10]\documentclass[12pt]{minimal}
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\begin{document}$$ b\in [2,10]$$\end{document}. This paper extends the previous works in Ddamulira et al. (Proc. Math. Sci. 127: 411–421, 2017) and Erazo et al. (J. Number Theory 203: 294–309, 2019).
