In a class speech situation, should a pupil ask their teacher:
Please Madam, can I come write the date (up) on the board?
Please Madam, can I come to write the date (up) on the board?
Additionally, is it better to use "up" in this request?
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You would say the first. It is implicitly a conjunction: Please, Madam, may I come (and) write the date? You are asking permission to come to the board, and to write the date. The test is that you could remove either verb and still have a grammatically correct sentence:
The second is a construction that means something different. In context, it means that you are asking whether one day you will become capable of writing the date.
To be complete, if you wanted to be absolutely unambiguous and insist that your coming to the board would only be for the express purpose of writing the date, you could say Please, Madam, may I come to the board in order to write the date? To my ear that sounds stilted, I would not expect a pupil to express themselves that way.
It is not necessary to add that you are writing up on the board, that is understood. In the request Please write that up on the board one says "up" to contrast with writing down on a sheet of paper.