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 $c \in \{6,7\}$.