Is financial assistance from the French government considered unlawful (haram) if the document submitted to obtain it includes a salary amount slightly less than the actual salary, and what should be done in this case?

It is permissible to receive assistance from the state if you meet the conditions and if the underestimated salary does not affect your eligibility or the amount of assistance. However, if it does affect the amount, then take only what you are entitled to and return the rest. If you are unable to return it, then donate it to the university or any other public entity.