Usually not. Paycheck withholding is sized by your W-4 as if the paycheck were your only income, so tax on side income lands on top of it unpaid.
Your employer withholds based on the W-4 you filed, and that form assumes the paycheck it is attached to is the whole picture. Side income arrives with nothing taken out. The tax on it stacks on top of a withholding amount that was never sized for it, and the gap shows up as a bill in April.
If the side income is self-employment, meaning freelancing, 1099 work, or selling things you make, it also owes self-employment tax of 15.3% on the profit. That applies on top of income tax and is the part that surprises most people with a day job.
There is a real chance your withholding covers it anyway, and the test is the safe harbor rule. If your withholding alone reaches 100% of last year's total tax, or 110% if your prior-year adjusted gross income was over $150,000, the IRS cannot penalize you no matter how much the side income adds. You would still owe the difference in April, but with no penalty. The same is true if you will owe less than $1,000 total.
If withholding will not get you there, you have two fixes. You can raise your paycheck withholding by putting an extra amount on line 4(c) of a new W-4, which is the simplest route because withholding counts as paid evenly through the year even if you raise it in November. Or you can make quarterly estimated payments on the side income directly.
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Sources: IRS Publication 505· Form W-4