What is the ruling of Sharia on my mother performing Hajj with money she received from the French state as widow's assistance, knowing that she secretly remarried less than a year after my father's death, and this money is her right as long as she is a widow, and she deprived me of my right to my father's inheritance?

It is not permissible for the mother to take money under the guise of being unmarried when she is not; she must return it. If she is unable to return it, she may spend it on public welfare. Her performance of Hajj with ill-gotten money is valid, though she incurs sin and her reward is diminished. As for the inheritance, if the grant was a gift, then the deceased has no relation to it, and it is rightfully owned by whomever the granting authority specifies. If it was from the entitlements of the deceased, then it is an estate to be divided among his heirs, and whoever was deprived of his share has the right to claim it.