Your employer is required to treat the reenlistment bonus as your wages.
Because bonuses are considered as supplemental wages - the income tax withholding is 25% - regardless how you filled W-4 form.
That is not an actual tax amount - the tax liability will be determine at the tax time based on your total income, deductions, credits, etc.
If your tax liability will be less than your withholding - you will claim a refund.
Let me know if you need any help.
If you file jointly with your wife and your expected gross income $88,000
assuming standard deduction, and that your child is below 17 (qualifies for child tax credit)
your taxable income is estimated as $65650
your income tax - ~$9000
child tax credit $1000
making work pay credit (if you both are working) $800
total tax - $7200
tax withholding from bonus - $13125
refund $5925 plus all withholdings from paychecks.
Please be aware that is a raw estimate.
Tax Preparer
Taxes, Immigration, Labor Relations