  In a previous work we computed the thermally-averaged cross sections for the
production and absorption of the $\chi_{c1}(4274)$ state in the hot hadron gas
formed in heavy ion collisions. In the present work we estimate the final yield
of this exotic state in these collisions. We use the coalescence model to fix
the initial multiplicities. The state is is treated as a $P-$wave bound state
of $D_s\bar D_{s0}$ and also as a compact tetraquark. The Bjorken picture is
used to model the hydrodynamic expansion and cooling. Then, the kinetic
equation is solved to evaluate the time evolution of the $\chi_{c1}(4274)$
yield during the hot hadron gas phase. Since the $\chi_{c1}(4274)$ decay width
is large it might decay inside the hadron gas. Therefore we also include the
$chi_{c1}(4274)$ decay and regeneration terms by means of an effective
coupling, estimated from the available data. The combined effects of hadronic
interactions and the $\chi_{c1}(4274)$ decay have a strong impact on the final
yield. Also, predictions of the $\chi_{c1}(4274)$ multiplicity as a function of
centrality and of the charged hadron multiplicity (measured at midrapidity) are
presented. Finally, we calculate the yield of a proposed $P-$wave molecular
state of $D_s \bar{D_{s0}}$, $Y^{\prime}(4274)$, characterized by a smaller
width and smaller coupling constant obtained from the Weinberg compositeness
condition.
