“clean up your act” = improve your behavior
“Hey. You’d better clean up your act, or Mom will throw you out of the house. You can’t be rude to her like that.”

“start off with a clean slate” = start things new again (children used to write on slates that could be erased)
“You had some problems here at home in the past, but I hope you can start off this year with a clean slate. Everyone needs a new beginning from time to time.”