Year end audit

If a certain dollar amount was charged to the stake but it should have been charged to a specific ward and the budget for the year has been closed, what is the best way to correct this error?

There is no way that will change the budget balances of the previous year. However, if the ward should have paid an expense but did not then they ended up with a larger rollover balance than they should have. Likewise the stake would have a smaller rollover. So if the ward makes a transfer to the stake then the current available budget on both sides will be corrected. But there will be a transfer transaction in this year and both the ward and stake may need to adjust the assigned budgets so that the amount available to be spent is correct.