How do I claim the $500 advance payment that was paid out to student last year

This is a $500 other dependent credit that a taxpayer can claim if the dependent is other than a qualifying child. here are the rules. For an example, many people provide support to their aging parents. But just because you mail your 78-year-old mother a check every once in a while doesn’t mean you can claim her as a dependent. Here is a checklist for determining whether your mom (or other relative) qualifies.

• Do they live with you? Your relative must live at your residence all year or be on the list of “relatives who do not live with you” in Publication 501. About 30 types of relatives are on this list.
• Do they make less than $4,300 in 2020 or 2021? Your relative cannot have a gross income of more than $4,300 in 2020 or 2021 and be claimed by you as a dependent.
• Do you financially support them? You must provide more than half of your relative’s total support each year.
• Are you the only person claiming them? This means you can’t claim the same person twice, once as a qualifying relative and again as a qualifying child. It also means you can’t claim a relative—say a cousin—if someone else, such as his parents, also claim him.

As far as claiming the dependent credit, you would enter these dependents in the my info section of your return. When you land on the summary page, look for the button that says "add a dependent."  From there, you add the dependent's name and answer the questions that are asked. From there, the Turbo Tax program will determine if this dependent qualifies to be yours.

You're describing the "Other dependent" credit.  It comes automatically, on your tax return, when you claim a dependent, other than a child eligible for the child tax credit.  It is a non-refundable credit. That is, it can only be used to reduce your tax liability.  It is not an advanced payment.

The other dependent credit replace the old dependent "exemption" deduction ($4000+/-) starting in 2018.