  The structure of $^{22}$C plays a vital role in the new physics at subshell
closure of $N=16$ in the neutron-rich region. We study the two-neutron
correlations in the ground state of the weakly-bound Borromean nucleus $^{22}$C
sitting at the edge of the neutron-drip line and its sensitivity to ${\rm
core}$-$n$ potential. For the present study, we employ a three-body (${\rm
core}+n+n$) structure model designed for describing the Borromean system by
explicit coupling of unbound continuum states of the subsystem (${\rm
core}+n$). We use a density-independent contact-delta interaction to describe
the neutron-neutron interaction and its strength is varied to fix the binding
energy. Along with the ground-state properties of $^{22}$C, we investigate its
electric-dipole and monopole responses, discussing the contribution of various
configurations. Our results indicate more configuration mixing as compared to
the previous studies in the ground state of $^{22}$C. However, they strongly
depend upon the choice of the $^{20}{\rm C}$-$n$ potential as well as the
binding energy of $^{22}$C, which call for new precise measurements for the
low-lying continuum structure of the binary system ($^{20}{\rm C}+n$) and the
mass of $^{22}$C. These measurements will be essential to understand the
Borromean three-body system $^{22}$C with more accuracy.
