Not an answer, but too long for a comment.
For each value of $k \ge 2$, there are several values of $n$ for which there is no such list. But there is an $N_k$ such, that every $n > N_k$, there is at least one such list (better called an integer partition of distinct $k$th powers). There are several entries in the Online Encyclopedia of Integer Sequences related to this, described below.
For $k = 2$, we're talking about possible sums of squares, a classical number theory topic. A003995 gives the $n$ values for which there is at least one sum of distinct squares equal to $n$; the largest $n$ not in the list is 128, so $N_2 = 128$. The smallest $n$ for which there are two sums of distinct squares is $n = 25$ from the Pythagorean triple $(3,4,5)$.
For $k = 3$, the analogous sequences are A003997 with the comment that $N_3 = 12{,}758$ and A003998 showing that $n = 216$ is the smallest that can be written as the sum of cubes in more than one way. (This list includes Hardy's "taxicab number" 1729.)
There are also OEIS entries for $n$ values with distinct $k$th power partitions for $k = 4, 5, 6$: respectively, A003999, A194768, and A194769.