  We investigate the effects of a disformal coupling between dark energy and
dark matter in the predictions of the spherical collapse and its signatures in
galaxy cluster number counts. We find that the disformal coupling has no
significant effects on spherical collapse at high redshifts, and in particular
during matter domination epoch. However, at lower redshifts, the extrapolated
linear density contrast at collapse close to redshift $z \lesssim 1$ and
overdensity at virialization can be strongly suppressed by a disformal coupling
between dark energy and dark matter. We also find that disformal coupling can
have different imprints on cluster number counts compared with conformal
coupling, such that the disformal coupling can strongly suppress the predicted
number of clusters per redshift interval at $z > 0.1$ while enhance the number
of cluster at $z < 0.05$. Using the specifications of eROSITA survey, we find
that the disformal coupling between dark energy and dark matter can be tightly
constrained by cluster number counts.
