The Estermann cubic problem with almost equal summands

We prove an asymptotic formula for the number of representations of a sufficiently large natural number N as the sum of two primes p1 and p2 and the cube of a natural numbermsatisfying the conditions |pi − N/3| ≤ H, |m3 − N/3| ≤ H, H ≥ N5/6ℒ10.