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?

Indeed it seems more complicated than that.  What if we take f the inclusion of a plane curve C into P^2 = Y, and X the blow up of P^2 at a point p of C.  Then the we seem to be able to find both constant and non constant models for h.  I.e. C' = C, and either h = id, and g is the "proper transform" of f,  or h is a constant map from C to the point p, and g is any non constant map to the exceptional P^1 in X.

So there seem to be various answers of various degrees corresponding to different positive dimensional components of the fiber product, and whether or not they surject onto f(C).  Have I messed this up?