Sherbourne was a boarding school; in schools like that the children often live in "houses". These may be separate buildings or just areas of the main building where they have dormitories, perhaps studies, and a common room. (Think Slytherin, Hufflepuff, and Griffindor etc. in Harry Potter). These houses would have responsibility for his pastoral care, so that section is reporting on that. I would say in this section dirty does indeed have the literal meaning, given that it's tied to "untidy".
I was a little more surprised to see the word used in the academic sections, but I still think it's the literal meaning, maybe referring to ink smudges or marks from mugs of tea or the like.
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