How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field?

I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the following situation:

Assuming that a variety V is defined by a polynomial H(z,t) over a field k(s), where k has finite characteristic, and s is transcendental, I'm especially interested in how many elements of form a+bs are contained in the Hilbert set of H, defined as the set of {r in k(s) such that H(z,r) is irreducible over k(s)}.

Any answers would be much appreciated!

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