So I was going to comment, instead of answer, but then I decided I wished to be a little more verbose.
Management insisting on reduced duplication in each sprint? Sign me up! I am usually fighting with them to allow us to spend time doing things like that, reducing technical debt sprint by sprint rather than simply piling it all up for some kind of bug-stomping fest, or god forbid, a massive production issue later!
You said sprint, so I am going to make a few assumptions here. I am a Certified Scrum Master, for what that may or may not be worth to you. So, Sprint: 1) you are using SCRUM or a similar framework. 2) You have planning meetings for each sprint, and commit to work in these planning meetings, after estimating it as a team.
You say you are pressed for time. Do you have an overall deadline that MUST happen, regardless of the team's commitment each sprint and how much they can deliver? If not, then when you estimate your stories at each sprint planning, try to take into account the cost of doing it 'right', IE not duplicating the logic in question, and instead wrapping it in some form of pattern that will make it reusable. Not only will you make management happy, but you will make yourselves happier as well! No more changing multiple copies of things when a business rule changes. No more wondering what where you have to go to find where you did it before, instead its part of an ongoing centralized library of tools at your disposal.
The key is: Nothing is free. If you have significant duplication (and I would say you do if even the management is seeing it as a problem), then explain that it takes time to fix that. The only way you have the time to fix it correctly is if they back off a bit on the time schedules, and let you do it. If you have control over commitment (you as in the developers) then commit to what you can do, realistically, and properly! If you don't have control over comitting to work (I have been there.) then talk to the PO or whomever DOES control the roadmap and release dating and explain the situation, the likely problems from not fixing it, and get them to agree.
If all of that fails, and management insists on duplication being reduced, but doesn't also guarantee the extra time that is needed to do it, you are between a rock and a hard place. The only thing the developer in me has for that is 'take the time you need to do it right'. I have never ever ever been reprimanded, by even my most unreasonable managers, for being late but getting it done right, over getting it done on time but having it blow up in a customers face. Sounds like now is the time to regroup and talk about it though, regardless.