Will a professor get annoyed with frequent requests for recommendation letters?
It is fine; don't worry about it.

This is a very common situation: many graduates will apply to more than one program that needs recommendation letters.  They almost certainly are going to be asking the same few people for multiple letters, because they are unlikely to have a large number of contacts to draw on to ask for recommendations.

I think the question you have linked very much applies to this case.  Particularly the top answer, which points out that this is part of professors' jobs, not some kind of exceptional favor you are asking for.