On an additive problem of unlike powers in short intervals

We prove that almost all positive even integers n can be represented as p22 + p33 + p44 + p55 with |pkk−14N|⩽N1−1/54+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {p_k^k - {1 \over 4}N} \right| \leqslant {N^{1 - 1/54 + \varepsilon }}$$\end{document} for 2 ⩽ k ⩽ 5. As a consequence, we show that each sufficiently large odd integer N can be written as p1 + p22 + p33 + p44 + p55 with |pkk−15N|⩽N1−1/54+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {p_k^k - {1 \over 5}N} \right| \leqslant {N^{1 - 1/54 + \varepsilon }}$$\end{document} for 1 ⩽ k ⩽ 5.