Representing an integer and its powers in two unrelated number systems

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation $$y^a=q_{N_1}+\cdots +q_{N_K},N_1\ge \cdots \ge N_K\ge 0,a,y\ge 2$$ where $(q_N{)}_{N\ge 0}$ is the sequence of convergent denominators to $\alpha$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca.