Title: A Diophantine approximation problem with two primes and one $k$-th power of a prime,
Abstract: We refine a result of the last two Authors of [8] 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 $1<k\le 3$ by
combining 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 common
range we also give a stronger bound for the approximation.