  The dimension free Harnack inequality is established for the distribution
dependent stochastic Hamiltonian system, where the drift is Lipschitz
continuous in the measure variable under the distance induced by the
H\"{o}lder-Dini continuous functions, which are $\beta
(\beta>\frac{2}{3})$-H\"{o}lder continuous on the degenerate component and
square root of Dini continuous on the non-degenerate one. The results extend
the existing ones in which the drift is Lipschitz continuous in the measure
variable under $L^2$-Wasserstein distance.
