A Cesàro average of generalised Hardy-Littlewood numbers

We continue our recent work on additive problems with prime summands: we already studied the average number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Cesaro weighted partial explicit formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results.