On the exceptional set of Hardy-Littlewood's numbers in short intervals

In 1923 Hardy and Littlewood conjectured that every sufficiently large integer is either a $k$-power of an integer or a sum of a prime and a $k$-power of an integer, for $k=2,3$. We will call HL-numbers the integers that are a sum of a prime and a $k$-power of an integer. Let now $kgeq 2$ and denote by $E_k$ the set of integers which are neither an HL-number nor a power of an integer. Here we prove that there exists an absolute positive constant $delta$ such that for $Hgeq X^{7/12(1-rac{1}{k})+delta}$ [ ert E_k(X,H)ert ll H^{1-delta/(5K)}, ] where $K=2^{k-2}$, thus improving previous results by Perelli-Pintz, Mikawa, Perelli-Zaccagnini and Zaccagnini.