If money is taken from an annuity to do home improvements, can it be written off?
You mean, made not taxable? No.
If you make improvements to your home, the cost is added to your cost basis, which may reduce the capital gains tax you owe when you sell. But improving your personal home is not a current tax deduction, and is not a reason to not pay tax on an annuity withdrawal that is otherwise taxable. (You are saving 4% interest you would pay if you took out an equity loan, so it might still be worth doing, depending on the APR of the annuity and your tax situation.)