Can I get some type of financial assistance or refund from TurboTax if I filed and then got another bill from the IRS for miscalculation and underpayment?

If it is an actual calculation error Turbo Tax pays any penalties but not the extra tax due.  But it is usually user error, like leaving off some income or entering an amount wrong or in the wrong place.  What did the IRS change?

Turbo Tax is a self prepared return.  It has no way of knowing or checking that you entered something wrong or in the wrong place or entered something twice  Or left something out.  It doesn't check against any outside sources.  The IRS has copies of all the W2s and 1099s your employer and banks/brokers, etc. sent in to match against.  Turbo Tax only checks the calculations of what you entered.   It is your responsibility to check it over before you file.  They only Guarantee that the calculations are right based on the entries you made.

https://ttlc.intuit.com/community/charges-and-fees/help/what-is-the-turbotax-100-accurate-calculation-guarantee/00/26355