From the math, you got a regular distribution of $626.71, and the plan withheld the mandatory 20% withholding of $125.34. You should get a 1099-R for this distribution, and the entire amount ($626) will be taxable, plus subject to a 10% early withdrawal penalty. Your total tax will be determined from this and all your other income. You will get credit for the $125 withholding along with all your other withholding and that will determine your overall refund or amount owed.
1. When did this happen (2019 or 2020)?
2. Why was only part of the money rolled over or transferred?
3. If this happened in 2019, what is the code in the 1099-R?
4. Did they put the money in an IRA that they are trustee for? (That's a direct institutional transfer which is similar to but not the same as a rollover.)
What did you do with the $501.37 check? The $1 25.34 was just an estimated amount for taxes.
You didn't actually pay the tax or 10% penalty (you pay a 10% early withdrawal penalty if you are under 59 ½). You had taxes withheld like from your paycheck. You still have to enter the whole amount (before taxes were withheld) with your other income to figure out the total tax (and it may put you into a higher tax bracket) and then the withholding is subtracted from the total tax to figure your refund or tax due. It has to break out and list the 10% early withdrawal penalty separately on your return. It's on 1040 Schedule 2 line 6. Then you will get credit for the withholding on 1040 line 17.