Does there exist a positive integer $n\in\mathbb{Z}$ and an irreducible polynomial $P\in\mathbb{Z}[X]$ of degree at least $2$ such that there are infinitely many pairs of positive integers $(m,k)$ with $P(m)=n^k$?
I have absolutely no idea how to attack the general problem, but for monic quadratic polynomials $P=X^2+aX+b$ it is logical to consider the discriminant of $P-n^k$, which is $$a^2-4(b-n^k)=4\cdot n^k+a^2-4b.$$ In order for $P$ to satisfy the condition, this discriminant has to be a perfect square for infinitely many $k$. For $k$ even and large enough this is impossible, because the discriminant is too close to (but not equal to) $(2n^{k/2})^2$. Therefore, if $P$ satisfies the condition, there must be infinitely many $l$ with $(2n^l)^2\cdot n+a^2-4b$ a perfect square.