Q. Is there any way to claim back all or part of that $5709 which came out of my paychecks?
A. Yes, to part. Almost certainly no to all.
The $5709 withholding is included in box 2 of your w-2 (unless that amount also include FICA) and will be credited to your calculated income tax.
If your coursework meets the criteria for "job related", your employer was not required to include it as imputed income (he treats it as job training, not education reimbursement). That's a determination they have to make. Once it's included on your W-2, you cannot claim it as a deduction, on your tax return. Your only chance is to get your employer to give you a corrected W-2.
