  The existence and uniqueness of mild solutions are proved for a class of
degenerate stochastic differential equations on Hilbert spaces where the drift
is Dini continuous in the component with noise and H\"older continuous of order
larger than $\ff 2 3$ in the other component.
  In the finite-dimensional case the Dini continuity is further weakened. The
main results are applied to solve second order stochastic systems driven by
space-time white noises.
