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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for all monic polynomials...