  In this paper, we consider Hankel operators, with locally integrable symbols,
densely defined on a family of Fock-type spaces whose weights are
$C^3$-logarithmic growth functions with mild smoothness conditions. It is shown
that a Hankel operator is bounded on such a Fock space if and only if its
symbol function has bounded distance to analytic functions BDA which is
initiated by Luecking(J. Funct. Anal. 110:247-271, 1992). We also characterize
the compactness and Schatten class membership of Hankel operators. Besides, we
give characterizations of the Schatten class membership of Toeplitz operators
with positive measure symbols for the small exponent $0<p<1$. Our proofs depend
strongly on the technique of H\"{o}mander's $L^2$ estimates for the
$\overline{\partial}$ operator and the decomposition theory of BDA spaces as
well as integral estimates involving the reproducing kernel.
