So let me check:
--There were computers used by students, while teachers are nearby Yes.
--The teachers, in the process of showing the students some things on the computer screens, inadvertently marked them with pencils Completely correct. Students also do this while showing things to teachers or other students.
--The "problem" is that after a while, the screens had pencil marks all over them No, but the marks are the cause of a problem.
Was the "puzzle" figuring out how to remove the pencil marks? No, but removing them is very relevant. how to prevent the marks from getting on them in the first place? No. trying to figure out how the marks got there (was this not known at first)? No, it is self-evident.
The students and teachers were at your workplace? Yes. do you do tech support at a school? Yes.college?
Does the "puzzle" part of the puzzle sentence pertain to a lateral thinking puzzle? Yes.