We can't possibly answer this question without the text to examine. However, I can try to point you in the right direction.
For the first part (tone), clear your mind, try to forget what you know about the story, and just read the letter. Do the words or ideas in it prompt any emotions? Those are your clues about the tone.
For the second part of the question ("how does she convince..."), you might try looking at the moment in the story before her mother lets her go, and the moment after. What changes between those two moments? Is her mother worn down by her persistence, for example, or does the girl use logical arguments that convince her mother?
Finally, at a guess, if the girl gets the sandwich at the canteen, it might represent freedom (i.e. the freedom to make her own lunch plans). However, I have no idea if this is right, since I don't know what story we're talking about.