Have the business people got burnt in the past by devs trying to clean the code? Do you have automated tests in place that make sure if you refactor the code that you can show it still functions in the same way?
The best way to handle this is in the course of normal feature work when you have to touch the code, that is when you clean it. So then you can just roll the change in with normal feature work and do not need to ask for the time to clean the code.
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