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Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,\\bfs{a})=\\mu_d.q+\\mathcal{O}(1), where 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},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the \\mathcal{O}--notation in terms of d and s with \"good\" behavior. 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