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We refine a result of the last two Authors of [8] on a Diophantineapproximation problem with two primes and a $k$-th power of a prime which wasonly proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ bycombining Harman's technique on the minor arc with a suitable estimate for the$L^4$-norm of the relevant exponential sum over primes $S_k$. In the commonrange we also give a stronger bound for the approximation.

A Diophantine approximation problem with two primes and one $k$-th power of a prime