  We investigate the effect of intermediate charmed meson loops on the M1
radiative decays $J/\psi \to \eta_c \gamma$ and
$\psi'\rightarrow\eta^{(\prime)}_c\gamma$ as well as the isospin violating
hadronic decays $\psi'\rightarrow J/\psi \,\pi^0(\eta)$ using heavy hadron
chiral perturbation theory (HH$\chi$PT). The calculations include tree level as
well as one loop diagrams and are compared to the latest data from CLEO and
BES-III. Our fit constrains the couplings of 1S and 2S charmonium multiplets to
charmed mesons, denoted $g_2$ and $g_2^\prime$, respectively. We find that
there are two sets of solutions for $g_2$ and $g_2^\prime$. One set, which
agrees with previous values of the product $g_2 g_2^\prime$ extracted from
analyses that consider only loop contributions to $\psi'\rightarrow J/\psi
\,\pi^0(\eta)$, can only fit data on radiative decays with fine-tuned
cancellations between tree level diagrams and loops in that process. The other
solution for $g_2$ and $g_2^\prime$ leads to couplings that are smaller by a
factor of 2.3. In this case tree level and loop contributions are of comparable
size and the numerical values of the tree level contributions to radiative
decays are consistent with estimates based on the quark model as well as
non-relativistic QCD (NRQCD). This result shows that tree level HH$\chi$PT
couplings are as important as the one loop graphs with charmed mesons in these
charmonium decays. The couplings $g_2$ and $g_2^\prime$ are also important for
the calculations of the decays of charmed meson bound states, such as the
X(3872), to conventional charmonia.
