For integer $k>3$, is something known about the rational points on

$$ \frac{ x^k-y^k }{ x-y } - (x-y)^{k-2} = 0 $$

It is genus 0 curve for $3 < k < 100$.

Coprime integer solutions are unlikely because of the Fermat-Catalan conjecture.

Rational solutions with coprime numerators and denominator of suitable size appear unlikely because of possible abc triples related to $x^k = y^k+(x-y)^{k-1}$. (example with large denominator for $k=5$ is the point $(\frac{2}{31}, \frac{1}{31})$ with abc relation $2^5=1^5+(2-1)^4 31$