  We investigate the neutral AdS black-hole solution in the consistent
$D\rightarrow4$ Einstein-Gauss-Bonnet gravity proposed in [K. Aoki, M.A. Gorji,
and S. Mukohyama, Phys. Lett. B {\bf 810}, 135843 (2020)] and construct the
gravity duals of ($2+1$)-dimensional superconductors with Gauss-Bonnet
corrections in the probe limit. We find that the curvature correction has a
more subtle effect on the scalar condensates in the s-wave superconductor in
($2+1$)-dimensions, which is different from the finding in the
higher-dimensional superconductors that the higher curvature correction makes
the scalar hair more difficult to be developed in the full parameter space.
However, in the p-wave case, we observe that the higher curvature correction
always makes it harder for the vector condensates to form in various
dimensions. Moreover, we note that the higher curvature correction results in
the larger deviation from the expected relation in the gap frequency
$\omega_g/T_c\approx 8$ in both ($2+1$)-dimensional s-wave and p-wave models.
