Hmm … haven’t tested that one experimentally, but, it might be because the mechanical stimulation caused by nose-picking irritates the mucosa which produces the mucopus (“bogeys”!).
Alternatively, it could be that chemicals (in dirt) on the inserted finger stimulates mucopus production.
Alternatively, it could be “association” rather than “causation”. This is an important thing to think about when testing hypotheses and scientific experiments. “Nose picking” and “more bogeys” might just happen at the same time (i.e. they are associated) but one might not cause the other. For example, it might be that allergens present in air in spring stimulate mucopus production and increase the frequency of picking (because the nose is irritated). Such an association can be hard to unpick … 😉
To see if there was causation, I was going to suggest a carefully controlled experiment, but I’m not sure that your teachers would be impressed!