One question on meaning of the phrase "be preoccupied with", I am not a native speaker and tried to find out the accurate meaning and nuance on the web search, but could not. I have seen that a native speaker use this phrase when he excused for a delayed response to a client as in "Sorry for my late feedback, we are preoccupied with many things in the past few weeks". I understand that this means "I have been busy for many things to do". But there is no any results relating to what I expected. Would my understanding on the meaning be correct?
Yes, you are correct. "to be preoccupied with something" means "to be busy with something" or "to be completely focussed on something".
The original sentence can be rephrased as: "Sorry for my late feedback, we have been busy with many things in the past few weeks"