We introduce a notion of "integral distance to analytic functions" in $\mathbb {C}^n$, denoted by $\mathrm{IDA}^{s,q}$, consisting of all locally $q$-integrable functions whose distance in $L^s$ to holomorphic functions is finite. Using $\mathrm{IDA}^{s,q}$ together with $\overline\partial$-techniques, we characterize all bounded (and all compact) Hankel operators $H_f$ from weighted Fock spaces $F^p(\varphi)$ to $L^q(\varphi)$. As an application, for bounded symbols $f$, with the help of the Calder\'on-Zygmund theory of singular integrals we prove that $H_f$ is compact from $F^p(\varphi)$ to $L^q(\varphi)$ if and only if $H_{\overline f}$ is compact, which generalizes Berger and Coburn's result on the classical (unweighted) Fock space $F^2$.