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In present work we have theoretically studied some features for generation of principally new type of radiation, so-called mixed Cherenkov-Channeling Radiation (ChCR). The earlier developed theory for relativistic electrons have been presently applied to the radiation emission by protons at medical accelerators. This type of radiation can be observed at protons channeling in optically transparent crystals, which is accompanied by channeling radiation (CR), and can be proposed as alternative to conventional Cherenkov radiation (ChR). We demonstrate that, as expected, ChCR photons are emitted at large angles with respect to the projectile momentum and close to the Cherenkov ones as well. It is shown that the ChCR intensity can essentially exceed the ChR one. Applying the numerical methods, the quantitative characteristics of ChCR for selected crystals as well as their distinctive peculiarities are analysed.