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I am probably confused. Your h seems always to be surjective. What if C = Y, f = id, and C = X, and phi = constant map to point p of C? Then we seem to need h to be constant. E.g. taking h = phi, and g = id, seems to work (and the minimum degree which occurs here is zero). Is it possible your construction (essentially) works as long as the images of f and phi meet? I.e. that it works more widely, but that it is only when the image of f is actually contained in the image of phi that h is surjective? No I guess you need the inverse image of f(C) in X, i.e. the fiber product, to contain a curve, in order for a non constant g to exist. Something like that?