Retirement tax questions

My point was, since the taxes were withheld, that portion that I am putting back in the IRA was already taxed.

 

No, that portion was NOT already taxed.  You only had withholding taken out.  That is not the actual tax.   You will get credit for the 20% withholding on your tax return and can get it back if all your withholding is more than  the tax due on all your income.