Should I put information about my repayment to TWC into my application if it's not mentioned in my 1099-G Form?

[ColeenD3]

That is correct.  In the year you make the a payments is when you will realize the deduction.  If you paid the benefits back in the same year, you can offset the amount of the repayments on your 1040. If you repaid the overpayment of unemployment benefits in the same year you received them,  subtract the amount of unemployment repayment from the total taxable amount you received. Report the difference as unemployment compensation.