Asymptotic formula in generalization of ternary Esterman problem with almost proportional summands

For $n \geq 3$, an asymptotic formula for the number of representations of a sufficiently large natural number $N$ in the form $p_1+p_2+m^n=N$, is obtained. Here $p_1$, $p_2$ are prime numbers, and $m$ is a natural number, satisfying the following conditions
$$ \left|p_k-\mu_kN\right|\le H, k=1,2, \left|m^n-\mu_3N\right|\le H, H\ge N^{1-\frac1{n(n-1)}}\mathscr{L}^{\frac{2^{n+1}}{n-1}+n-1}. $$