  We study the chemistry of small hydrocarbons in the photon-dominated regions
(PDRs) associated with the ultra-compact HII region Mon R2. Our goal is to
determine the variations of the abundance of small hydrocarbons in a high-UV
irradiated PDR and investigate their chemistry. We present an observational
study of CH, CCH and c-C$_3$H$_2$ in Mon R2 combining data obtained with the
IRAM 30m telescope and Herschel. We determine the column densities of these
species, and compare their spatial distributions with that of polycyclic
aromatic hydrocarbon (PAH). We compare the observational results with different
chemical models to explore the relative importance of gas-phase, grain-surface
and time-dependent chemistry in these environments. The emission of the small
hydrocarbons show different patterns. The CCH emission is extended while CH and
c-C$_3$H$_2$ are concentrated towards the more illuminated layers of the PDR.
The ratio of the column densities of c-C$_3$H$_2$ and CCH shows spatial
variations up to a factor of a few, increasing from
$N(c-C$_3$H$_2$)/N(CCH)\approx0.004$ in the envelope to a maximum of
$\sim0.015-0.029$ towards the 8$\mu$m emission peak. Comparing these results
with other galactic PDRs, we find that the abundance of CCH is quite constant
over a wide range of G$_0$, whereas the abundance of c-C$_3$H$_2$ is higher in
low-UV PDRs. In Mon R2, the gas-phase steady-state chemistry can account
relatively well for the abundances of CH and CCH in the most exposed layers of
the PDR, but falls short by a factor of 10 to reproduce c-C$_3$H$_2$.
  In the molecular envelope, time-dependent effects and grain surface chemistry
play a dominant role in determining the hydrocarbons abundances. Our study
shows that CCH and c-C$_3$H$_2$ present a complex chemistry in which UV
photons, grain-surface chemistry and time dependent effects contribute to
determine their abundances.
