The paper under review establishes the following two theorems about simultaneous representations of pairs of positive integers as sums of prime powers and powers of 2. Theorem 1. For \(k_1=34\), the system of equations \[ \begin{split} N_1 &= p_1+p_2^2+p_3^3+p_4^4 +2^{v_1} +2^{v_2} + \cdots +2^{v_{k_1}},\\ N_2 &= p_5+p_6^2+ p_7^3+ p_8^4+2^{v_1}+2^{v_2} +\cdots +2^{v_{k_1}} \end{split} \] is solvable for every pair of sufficiently large even integers \(N_1\) and \(N_2\) with \(N_1\asymp N_2\). Theorem 2. For \(k_2 = 36\), the system of equations \[ \begin{split} N_1 &= p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4+2^{v_1} +2^{v_2} + \cdots +2^{v_{k_2}},\\ N_2 &= p_7^2+p_8^2+p_9^3+p_{10}^3+p_{11}^4+p_{12}^4+ 2^{v_1} +2^{v_2} +\cdots +2^{v_{k_2}} \end{split} \] is solvable for every pair of sufficiently large even integers \(N_1\) and \(N_2\) with \(N_1\asymp N_2\). The proofs rely on the circle method. They require establishing accurate numerical estimates for the singular series and integral.