Is the income earned by the husband from part-time work while continuing to receive government financial assistance without disclosing his real income considered unlawful, and what is the wife's stance on that?

These grants and aids are only deserved if the conditions set by the granting authority are met. If the state, had it known the salary your husband earns from his work, would not have given him the aid, then this aid is not permissible for him.

You must command your husband to be truthful and to seek lawful earnings.

As for your benefiting from his money for sustenance and other needs, it is permissible as long as he has a lawful source of income from his work; for consuming mixed (lawful and unlawful) money is permissible.