  In this paper, we investigate exceptional sets in the Waring-Goldbach problem
for unlike powers. For example, estimates are obtained for sufficiently large
integers below a parameter subject to the necessary local conditions that do
not have a representation as the sum of a square of prime, a cube of prime and
a sixth power of prime and a $k$-th power of prime. These results improve the
recent result due to Br\"udern in the order of magnitude. Furthermore, the
method can be also applied to the similar estimates for the exceptional sets
for Waring-Goldbach problem for unlike powers.
