The distribution of divisors of polynomials

Let \(F(x)\) be an irreducible polynomial with integer coefficients and degree at least 2. For \(x\ge z\ge y\ge 2\), denote by \(H_F(x, y, z)\) the number of integers \(n\le x\) such that \(F(n)\) has at least one divisor \(d\) with \(y0\) is arbitrarily small.