All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
If this mistake is significant and critical in relation to receiving the social welfare, in the sense that had you said the truth, you would have been denied the welfare, then you have to return the money to the concerned authority, even if indirectly, as you are not entitled to it. The Prophet, sallallahu ‘alayhi wa sallam, said, “The hand that takes is responsible for what it has taken until it returns it.” [At-Tirmithi] Your action falls under prohibited lying and making false statements. Allah, The Exalted, says (what means): {...and avoid false statement...} [Quran 33:30] Ar-Raaghib said that the "false statement" in the verse refers to lying.
Al-Haafith Ibn Hajar wrote, “False means describing something contrary to its actual truth. It may refer to a statement indicating lying, or a false testimony, or a false action ...” [Fat-h Al-Baari]
Hence, you should seek forgiveness from Allah, The Exalted, repent to him of this act, and resolve never to do it again. Also, return what you have received, and you may use the things that you bought with that money.
If you cannot return the money even indirectly, then you should pay it to the poor and needy. If you are poor, then you may take it yourself. An-Nawawi wrote, “If the unlawful wealth is offered to a poor person, it is not deemed unlawful for them; instead, it is lawful and pure for the poor to spend. The person whose wealth or income is acquired through unlawful means may give his (unlawful) money as charity to himself or his dependents if he is poor; rather, they deserve his charity more than anyone else; and he may take what is proportionate to his need because he is also poor.” [Al-Majmoo’]
Allah knows best.