Have you ever been asked to implement a feature with the first question being 'how long will it take?' for you to pluck a figure out of the air?
In these scenarios a voice inside my head wants to ask 'well, you are the manager, aren't managers supposed to know these things?'
There is an aspect of this going on here. The manager should have some idea of what is involved in the task and have an idea as to how best to solve the task.
Once upon a time I learned a lesson about lying from The Simpsons:
Homer: Marge, it takes two to lie: one to lie and one to listen.
There is an aspect of that to this situation. I also am not too sure that the deliberate errors are the wrong thing to do. With my coding errors I often jest that these are 'deliberate mistakes' there to just see if anyone is paying attention. I have also joked before now that there is 'one deliberate mistake' in there to check that people are doing the testing properly.
In a fake-news world where people can be economical with the truth the OP has the option to weasel out of the deception that way.
So how to rescue the situation in such a way that everyone wins?
Maybe the OP can taper down the deliberate errors to make the work perfect. If there is also a claimed drop in 'time taken' to do this and some vague talk of 'macros' used to 'double check the work' then a slightly more honest bargain can be struck where it is acknowledged that some initiative has been taken to do things better is rewarded and not punished.
For the line manager this can be a definite win with nobody apparently played as a fool.