Exceptional set in Waring–Goldbach problem for sums of one square and five cubes

Let $ N $ be a sufficiently large integer. In this paper, it is proved that, with at most $ O\big(N^{4/9+\varepsilon}\big) $ exceptions, all even positive integers up to $ N $ can be represented in the form $ p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3 $, where $ p_1, p_2, p_3, p_4, p_5, p_6 $ are prime numbers.