Small prime kth power residues

Let k\geq 2 be an integer. Let \epsilon >0 be given. It is widely believed that the smallest prime that is a kth power residue modulo a prime q should be O(q^{\epsilon }) for any \epsilon >0. Elliott has proved that for large primes q\equiv 1\bmod k such a prime is at most q^{\frac {k-1}{4}+\epsilon } for each \epsilon >0. We show that for large primes q\equiv 1\bmod k, the number of primes p\leq q^{\frac {k-1}{4}+\epsilon } such that p is a kth power residue \bmod \, q is at least q^{\frac {9\epsilon }{20k}}.