Someone correct me if I'm mistaken (I'm new to Latin), but it is my understanding that the classical Latin pronunciation of "c" is always "k". In later ecclesiastical Latin, "c+e" and "c+i" would become "ch+e" and "ch+i", respectively. It remains that way in Italian, and a similar effect is found in Spanish, but with those combinations becoming "th+e" and "th+i" in Spain, and "s+e" and "s+i" in Mexico.
You're right, in Classical Roman Latin the letter C was always pronounced with a hard /k/ sound. The Romance Languages went through phonetic changes which made so that, nowadays, the letter C has a soft /s/ sound in combination with E and I in Portuguese and Latin American dialects of Spanish, for example. Ecclesiatical Latin has that /t͡ʃ/ ("ch" sound) before E and I because it's based on Italian phonetics.