How to incentivise efficiency in a government?

One of the main aspects of most economic schools is the inefficiency of government with the base underlying reason being close to them not being a profit-oriented but power/popularity-oriented organisations.

I have little understanding of budgeting and know that there are ways to encourage efficiency in projects (with monetary benefits on a complex basis of responsibility per each task) but I don’t know if there are any similar approaches for budgets that don’t have a particular goal at the end.

Then, I also thought about a particular question. Let’s assume a local government structure somehow completed all their yearly goals and also got some spare money from the budget at the end of the year. What should they do with the saved portion of the budget?

Should they give a refund to their relevant tax payers? Should they invest it? Reduce taxes somehow? Should they give bonuses to the decision makers or to the general governmental employees? Will this be easily abused?