So the applicant is a school.
If your invention has nothing to do with the school, then the applicant should be yourself and the school has no right to apply.
Back to this topic, it depends on what kind of patent funding. Usually, it subsidizes the patent application fee, agency fee or other official expenses, all of which are paid by the school.
There is also a government incentive fee. If it is a reward for an invention, it depends on whether he rewards the inventor. But generally speaking, there are few patents granted to inventors, all of which are for applicants.