A Note on Waring–Goldbach Problem: One Square, Four Cubes and One

This paper establishes that for k ≥ 4, all sufficiently large even integers n ≤ N, with at most O(N1/2 − ϑ(k) + ε) exceptions, admit representations as the sum of one square of a prime, four cubes of primes and one kth power of a prime, where the exponent ϑ(k) relies on k. This result sharpens the bound previously obtained by [J. J. Li, F. Xue and M. Zhang, Bull. Aust. Math. Soc. 107 (2023) 416–431].