You cannot make a payment through TurboTax nor can TurboTax contact the IRS on your behalf.
If you agree with the IRS notice then you can pay the federal taxes owed on the IRS payment website using Direct Pay - https://www.irs.gov/payments
If you disagree with the IRS then follow the procedure in the IRS notice you recieved.
IRS website for a CP14 notice - https://www.irs.gov/individuals/understanding-your-cp14-notice
There was just another post about CP14 letter. It was for "when you don't pay enough taxes due for the year with your quarterly estimated tax payments, we charge a penalty for not properly estimating your tax."
Does yours say that too? If you had a tax due, It's common and expected for the IRS to send a bill for an underpayment penalty. It is determined by when the IRS actually gets your tax due payment which Turbo Tax won't know.
Or how did you pay the tax due? Were you suppose to send a check or the IRS pull it out of your account? Do you have proof it got paid? Maybe the IRS had trouble debiting your bank account.