On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k}}, \hfill \\ {\left| {{p_j} - \sqrt {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)