Exceptional sets in Waring-Goldbach problem for fifth powers

We consider exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we prove that all but O(N^131/132) integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes, which improves the previous results due to A. V. Kumchev Canad. J. Math., 2005, 57: 298–327] and Z. X. Liu Int. J. Number Theory, 2012, 8: 1247–1256].