Slim exceptional sets of Waring-Goldbach problem: two squares, two cubes and two biquadrates

Let $N$ be a sufficiently large number. We show that, with at most $O(N^{3/32+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$, where $p_1, p_2, …, p_6$ are prime numbers. This is an improvement of the result $O(N^{7/18+\varepsilon})$ due to Zhang and Li. Bibliography: 13 titles.