It's too late to have tax withheld from unemployment benefits that you have already received. You can look into having tax withheld from future unemployment benefits. Depending on your state, you might be able to have extra tax withheld to make up for not having had tax withheld on the extra $600.
In some states (California is one), if you ask to have tax withheld from your unemployment benefits, they will withhold tax only on the regular state unemployment amount, not on the additional amount, which comes from the federal government. In such cases, the withholding will probably not be enough to cover the tax on the extra federal payment. The additional $600 from the federal government ended the last week of July. There are attempts underway to extend it, but probably not at the same amount.
Your unemployment benefits might also be taxable on your state income tax return. Some states tax unemployment and some don't. Check the rules for your state. If it is taxable, the entire amount, including the extra $600, will be taxable, so you might need to adjust your state withholding for the rest of the year, or make state estimated tax payments. All unemployment benefits are always taxable on your federal tax return.
