I received a lesser amount of my refund than the irs had stated what my refund would be after they / the irs intercepted partial of the refund to pay back taxes.

if the IRS took part of your refund for back taxes you can call the following number to get info. You should also receive in the mail explanation

The federal Bureau of the Fiscal Service (BFS), part of the Department of the Treasury, issues refunds and tracks offsets. Call the Bureau of the Fiscal Service directly at this toll-free number: (800) 304-3107.