Does Higgs Boson influences with Gravity? Is there any way that if the universe is expanding Higgs particle has influence over it?
The ubiquitous "nonzero vacuum expectation value" of the Higgs field, by which the Higgs supplies mass to quarks, leptons, and W and Z bosons, should actually be contributing to the vacuum energy of the universe, and therefore to the dark energy, in a way that isn't seen. So there is definitely something we don't understand about how the Higgs field interacts with gravity, e.g. perhaps something unknown cancels out its contribution to the vacuum energy. This is one aspect of what is called the cosmological constant problem.
In addition to the V.E.V. of the Higgs contributing to the vacuum energy (part of the cosmological constant problem; there'a s good review in Rev. Mod. Phys. by Weinberg from the '80s), there have been investigations that the Higgs could be the inflaton field, which drove inflation in the early universe.
However, I think such a model is disfavoured, but I haven't looked into it.
Search `Higgs inflation Shaposhnikov' (the main proponent is M. Shaposhnikov) in Google.