I was asked for an estimate for a project I was working on. I said ~65 days. Now the deadline will give me around 40 days of work on it. I'll need to cut corners to make that, but I know in the end it won't be a finished product that we will be releasing and it will cost more time in the long run as I'll need to go back and redo parts of it. But we have a deadline now so I'll need to cut corners. The obvious way to do that is to focus on the most used parts of the application being done well, and finish / refactor the less used parts later.

Well, no, I don't agree. There are lots of opportunities to say "this really ought to be refactored... but whatever, it more or less works" when working on a typical application.