Due to a payroll error at work, my HSA contribution got put in twice for the first pay week. Once pre-tax and once post-tax. The extra total amount being in there is fine as it won't cause any limit issues. However, I've paid tax on money I should not have. Is there a way to account for/correct that come tax time or does Uncle Sam just get a small bonus from me for this year?
When you contribute money "pre-tax" into your HSA from your paycheck, you avoid payroll tax (FICA) on it. When it comes to tax time next year, that amount is not included on your W-2 box 1 amount, so you don't need to deduct it on your income taxes, because it was already automatically deducted from your income ahead of time.
However, with a "post-tax" HSA contribution, you paid payroll tax on that money before it went into your HSA. Also, that amount will be included on your W-2 box 1, so in order for you to get the tax benefit from the HSA contribution, you need to deduct it from your income on your tax return. This is done with Form 8889. On that form, the amount that you contributed "post-tax" should be put on line 2, and the amount that was contributed "pre-tax" out of your paycheck should be included on line 9.
By doing this, the "post-tax" contribution will become an income tax deduction, so you ultimately won't pay income tax on that amount. However, the payroll tax you paid on it is not refundable.
When you get your W-2 next year for 2019, you should confirm that everything is as you think it should be. Your Box 1 income should not include the "pre-tax" contributions, but should include the "post-tax" contributions.
There's not really such a thing as "pre-tax" or "post-tax" money, it's all fungible. The only thing that matters is the amount of money in total, and the total amount that's taxable. The latter calculation absolutely can and should be corrected by your payroll provider, resulting in you paying the correct amount of tax in the end (even if the original withholding was wrong).
Insist that your payroll provider correct this in their records before they generate your W-2... it's very risky to try to make a correction yourself on your tax return because your numbers won't match the numbers the IRS got from the W-2 copy sent to them.