Let $m\ge 1$ be an integer.
For which $m$ can we find odd perfect powers $a,b$ with $a+b=2^m$ ? The only solution, I found for $m\le 80$ is $m=9$ with the representation $$2^9=13^2+7^3$$
The exponents cannot be both even because the sum of two odd squares is congruent $2$ modulo $4$. Can we prove that no other power of $2$ does the job ? If the question is too difficult, can at least the case $p^2+q^3$ be solved with elliptic-curve-properties ?