Goldbach-Linnik type problems for mixed powers of primes and powers of 2

Let $p_1,p_2,\dots ,p_7$ be primes. In this paper, we first show that when $k_1=23$, every sufficiently large odd integer can be represented as the sum of one prime square, five prime cubes, one prime biquadrate and at most $k_1$ powers of $2$. We further prove that for $k_2=45$, every pair of sufficiently large odd integers satisfying certain necessary conditions can be represented as a pair of equations involving one prime square, five prime cubes, one prime biquadrate and at most $k_2$ powers of $2$.