Let $P_n(x) = \sum_{k=0}^{n} ξ_k x^k$ be a Kac random polynomial, where the coefficients $ξ_k$ are i.i.d.\ copies of a given random variable $ξ$. Based on numerical experiments, it has been conjectured that if $ξ$ has mean zero, unit variance, and a finite $(2+\varepsilon_0)$-moment for some $\varepsilon_0>0$, then \[ \mathbb{E}[N_{\mathbb{R}}(P_n)] \;=\; \frac{2}π \log n + C_ξ + o_n(1), \] where $N_{\mathbb{R}}(P_n)$ denotes the number of real roots of $P_n$, and $C_ξ$ is an absolute constant depending only on $ξ$, which is nonuniversal. Prior to this work, the existence of $C_ξ$ had only been established by Do-Nguyen-Vu (2015, \emph{Proc.\ Lond.\ Math.\ Soc.}) under the additional assumption that $ξ$ either admits a $(1+p)$-integrable density or is uniformly distributed on $\{\pm 1, \pm 2, \dots, \pm N\}$. In this paper, using a different method, we remove these extra conditions on $ξ$, and extend the result to the setting where the $ξ_k$ are independent but not necessarily identically distributed. Moreover, this proof strategy provides an alternative description of the constant $C_ξ$, and this new perspective serves as the key ingredient in establishing that $C_ξ$ depends continuously on the distribution of $ξ$.