Representing integers as sums of mixed powers of primes

We establish two new Waring–Goldbach type representations: every sufficiently large odd integer $n$ can be expressed as $$n=p_1^2+p_2^2+p_3^3+p_4^3+p_5^5+p_6^6+p_7^c,$$ where each $p_i$ is prime and $c\in \{6,7\}$.