On the Waring-Goldbach problem for one square and five cubes in short intervals

summary:Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$ n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3, \Bigl | p_1^2-\dfrac {N}{6}\Bigr | \leq U, \quad \Bigl | p_i^3-\dfrac {N}{6}\Bigr |\leq U, \quad i=2,3,4,5,6, $$ where $U=N^{1-\delta +\varepsilon }$ with $\delta \leq 8/225$