Actually, yes. It can be evidence of presence. Also, depending on the benefit applied for and received and whether or not it was a benefit to a US Citizen or non-citizen, this can raise questions about public charge. Please see an immigration attorney so a professional can analyze your individual situation.
No, because it is not a disqualifier for it unless you lie on the forms.
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I agree with Ms. Ramos, yes it can be evidence of presence. I'm not quite sure what the circumstances surrounding your situation are, and I would encourage you to seek the help of an immigration professional to assess your situation. Are you trying to prove presence or are you trying to avoid presence? Either way, it is always best to be honest and again, ask for professional assistance. As far a public charge, I'm not sure what Mr. Segal is getting at, but yes there can be a problem with certain applications if you or your children were or are on public assistance.
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A public charge issue might be raised by USCIS. It is best to consult an attorney.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. Consult with a qualified attorney before making any legal decisions. Gen Kimura, (832) 247-6932.