We have studied the dominant radiative transitions of the charmonium $S$- and $P$-wave states within the CCQM. The gauge invariant leading-order transition amplitudes have been expressed by using either the conventional Lorentz structures, or the helicity amplitudes, where it was effective. The renormalization couplings of the charmonium states have been strictly fixed by the compositeness conditions that excludes the constituent degrees of freedom from the space of physical states. We use the basic model parameters for the constituent c-quark mass $m_c=1.80$ GeV and the global infrared cutoff $lambda=0.181$ GeV. We additionally introduce only one adjustable parameter $varrho>0$ common for the the charmonium states $eta_c({}^1!S_0)$, $J/psi({}^3!S_1)$, $chi_{c0}({^{3}}!P_{0})$, $chi_{c1}({^{3}}!P_{1})$, $h_c({^{1}}!P_{1})$, and $chi_{c2}({^{3}}!P_{2})$ to describe the quark distribution inside the hadron. This parameter describes the ratio between the charmonium size and its physical mass. The optimal value $varrho=0.485$ has been fixed by fitting the latest data for the partial widths of the one-photon radiative decays of the triplet $chi_{cJ}({^{3}}!P_{J}),~J={0,1,2}$. Then, we calculate corresponding fractional widths for states $J/psi({}^3!S_1)$ and $h_c({^{1}}!P_{1})$. Estimated results are in good agreement with the latest data. By using the fraction data from PDG2020 and our estimated partial decay width for $h_c({^{1}}!P_{1})$ we recalculate the theoretical full width $Gamma^{rm theor}_{h_c} simeq ( 0.57 pm 0.12 )$ MeV in comparison with latest data $Gamma^{rm exp}_{h_c} simeq (0.7pm 0.4)$ MeV. We also repeated our calculations by gradually decreasing the global cutoff parameter and revealed that the results do not change for any $lambda<0.181$ GeV up to the deconfinement limit.
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