This is very similar to the Waring-Goldbach problem, which there are quite a few papers on (primarily when the $\alpha_i$'s are all equal but there are some results on mixed exponents), which asks how large $n$ must be in order to guarantee solutions to
$$p_1^\alpha + \cdots + p_n^\alpha = N,$$
for all sufficiently large $N$. In other words, there's nothing special about the RHS being a prime power: once you have a large enough (finite) number of terms, you can generate any number by summing prime powers together.
If we relax your requirement that the primes be distinct, then there are many solutions when $n$ is sufficiently large compared to $\alpha$. This web page suggests there are asymptotics available for $n \gg \alpha^2 \log \alpha$, presumably via the circle method. If so, then it's highly likely one could incorporate your restriction that the primes are all distinct: the asymptotics for the number of solutions where certain primes are equal will be much smaller when $N$ is very large compared to $n$.