Find all pairs of positive integers $(n, k)$ such that all sufficiently large odd positive integers $m$ are representable as $$m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}$$for some non-negative integers $a_1, a_2, \ldots, a_{k+1}$.
Take some large number $N$. Then the number of possible values of $a_1$ to represent $m \le N$ is clearly only $M^{\frac{1}{n^2}}$, the number of $a_2$ is $M^{\frac{1}{(n+1)^2}}$ etc. So the total number is at most $M^{\frac{1}{n^2}+\frac{1}{(n+1)^2}+\dots+\frac{1}{(n+k)^2}}$.
So we clearly need the exponent $\frac{1}{n^2}+\frac{1}{(n+1)^2}+\dots+\frac{1}{(n+k)^2} \ge 1$ to cover all sufficiently large odd $m$.
But this means that $n=1$ since otherwise the sum is at most $\frac{\pi^2}{6}-1<1$.
On the other hand, if $n=1$, then any $k$ works since we can just choose $a_1=m$ and the rest to be zero.
So the answer is all pairs $(1,k)$.