Prime powers in sums of terms of binary recurrence sequences

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Let $$(u_{n})_{n \ge 0}$$ be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation $$u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$$ with $$n_1> n_2> \cdots > n_t\ge 0$$ . Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő.