Some simple observations, which are independent of x or y being prime powers.

m must be less than log N. If there is a solution, N has to factor into (about) d(m) factors which are values of cyclotomic polynomials of indices c dividing m, and thus the sizes of the factors are close to x^phi(c) in size, so one can't have just any product of factors.

We have m=2 is always a solution with x = N-1, so if x must be a prime power then so is N-1. For N with few factors, one can rule out m with many divisors. Large prime factors of N must exist when m is greater than 6, and these primes must be 1 mod m, so if there are two of the prime factors of N greater than log N, their difference can give some restrictions on the possible values of m.

Ribenboim's book on Catalan's conjecture will have related material, and possibly a good answer to your question.

Gerhard "Playing Around With Prime Numbers" Paseman, 2018.10.05.