No. The taxable and nontaxable portions of an annuity distribution are determined by law.
Even if you do report a higher amount as taxable than the amount reported in box 2a of the Form 1099-R from the annuity company, it will not change the fact that the legally required amount of your investment in the contract was actually distributed and will not cause more of your investment in the contract to remain in the annuity. The taxable amount of future distributions from the annuity will not be reduced by you incorrectly reporting the taxable amount. The IRS automated system might or might not detect your reporting error, but if the inappropriate increase in your AGI it doesn't somehow increase your refund associated with the tax return on which you made the error you might not hear anything about it from the IRS.
