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

We refine a result of the last two Authors on a Diophantine approximation problem with two primes and a k-th power of a prime which was only proved to hold for 1<k<4/3. We improve the k-range to 1<k≤3 by combining Harman's technique on the minor arc with a suitable estimate for the L4-norm of the relevant exponential sum over primes Sk. In the common range we also give a stronger bound for the approximation.