  We obtain an estimate on the average cardinality of the value set of any
family of monic polynomials of Fq[T] of degree d for which s consecutive
coefficients a_{d-1},...,a_{d-s} are fixed. Our estimate asserts that
\mathcal{V}(d,s,\bfs{a})=\mu_d\,q+\mathcal{O}(q^{1/2}), where
\mathcal{V}(d,s,\bfs{a}) is such an average cardinality,
\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},...,a_{d-s}). We
also prove that \mathcal{V}_2(d,s,\bfs{a})=\mu_d^2\,q^2+\mathcal{O}(q^{3/2}),
where that \mathcal{V}_2(d,s,\bfs{a}) is the average second moment on any
family of monic polynomials of Fq[T] of degree d with s consecutive
coefficients fixed as above. Finally, we show that
\mathcal{V}_2(d,0)=\mu_d^2\,q^2+\mathcal{O}(q), where \mathcal{V}_2(d,0)
denotes the average second moment of all monic polynomials in Fq[T] of degree d
with f(0)=0. All our estimates hold for fields of characteristic p>2 and
provide explicit upper bounds for the constants underlying the
\mathcal{O}--notation in terms of d and s with "good" behavior. Our approach
reduces the questions to estimate the number of Fq--rational points with
pairwise--distinct coordinates of a certain family of complete intersections
defined over Fq. A critical point for our results is an analysis of the
singular locus of the varieties under consideration, which allows to obtain
rather precise estimates on the corresponding number of Fq--rational points.
