An average number of square-free values of polynomials

The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Usually, it is more difficult to obtain a similar order of error term for a higher degree polynomial $f(x)$ in place of $\text{Id}(x)$. Under the Riemann hypothesis, we show that the error term, on average in a weak sense, over polynomials of arbitrary degree, is of the expected order $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$.