Your refund is based on your total income and the total amount of tax that you paid or had withheld during the year. It's not calculated separately for specific items of income. The amount withheld from the annuity distribution is added to your total withholding. It is not applied specifically to the distribution.
Additional income increases your Adjusted Gross Income (AGI). That can sometimes have side effects. It can reduce or eliminate unrelated deductions or credits. If you receive Social Security benefits, the additional income can make more of your Social Security taxable. It can also push you into Net Investment Income Tax or increase your Net Investment Income Tax if you are already subject to it. Also, the refund "meter" at the top of the TurboTax screen is not really accurate until you have entered all of your information.
To figure out why your refund was reduced you would have to compare your Form 1040 line by line before and after entering the annuity distribution and see what else changed. Before you do that, double-check your 1099-R entries to make sure there isn't an error in the entries.
