Except in the case where the annuity is an IRA annuity, there is nothing you can do to avoid including in income the taxable amount of the annuity distributions.
If the annuity is an IRA annuity and you are over age 70½ at the time of the distribution, you can have distributions (up to $100,000 annually) paid by the IRA custodian directly from the IRA annuity to charities as Qualified Charitable Distributions. QCDs are not includible in income.
