Let $f(x,y)=0$ be irreducible elliptic curve over the rationals.

Are there $f$ for which:

Both $x,y$ are arbitrary large powers infinitely often, i.e. infinitely many rational points $(u^k,v^m)$ with both $k,m$ arbitrary large?

For $x,y$ squares (asked in previous revision) this is possible. Take $f(x,y)=x^{6} - 2 x^{3} y^{3} + y^{6} - 72 x^{3} - 72 y^{3} + 1296$.

$f(x^2,y^2)$ is divisible by $x^3 + y^3 - 6$ which is genus $1$ of positive rank.

For large $k,m$ the n-conjecture implies the number of monomials can't be too small.