Can I file Form 843 and not pay the penalty when I make my payment or do I have to pay the tax + penalty owed on Form 1040 and receive a refund via Form 843?

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TurboTax does support IRS Form 843

IRS Form 843, Claim for Refund and Request for Abatement - https://www.irs.gov/pub/irs-pdf/f843.pdf

IRS Form 843 instructions - https://www.irs.gov/pub/irs-pdf/i843.pdf