Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes

Let $p_1,p_2,\dots ,p_6$ be prime numbers. First we show that, with at most $O\left(N^{1/12+\epsilon }\right)$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_1^2+p_2^2+p_3^3+p_4^3+p_5^3+p_6^3$, which improves the previous result $O\left(N^{1/4+\epsilon }\right)$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O\left(N^{5/12+\epsilon }\right)$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3$.