Last week I received a 'Notice of Additional Tax Due' from New York State stating that I failed to report Federal audit changes to them. This was for the tax year 2010 and stated I owe an additional $211.
Of course I didn't remember any audit changes and pulled all my records for that tax year. Surprise! I found a notice from the IRS dated 2/14/2012 that I owed an additional $625 (plus interest) because I understated my wife's gambling winnings by $2,500. We paid the additional tax at the time but for some reason I neglected to dig deeper into our records.

A little background:
The reason for the understatement was because my wife had received two receipts from the NYS lottery for $2,500 in winnings. Each receipt was the same date, same amount and except for one digit in the receipt number, identical in every respect. At the time, I thought one was a copy for my records and only reported $2,500 plus $1,300 in slot winnings from Caesars in Atlantic City.
We itemized our deductions for that year and she had gambling losses exceeding her winnings (I have a win/loss statement for that year from Caesars).
We never should have paid the additional taxes because her losses still exceeded her winnings even with the additional $2,500.

The question:
Is there any way I can have the IRS refund the extra $625 we paid in error and do I have to pay the additional money New York State says we owe them?

Thank you in advance for your help.