2. Degree of irrationality and covering gonality

Problem 2.1.
Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$? How do we make sense of these notions over countable / finite fields?
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.