Don't know what's a Chingam? Let's make it easier. This thing is also known as chewing gum in most places which are not villages and small towns of India, or should I say even Indian subcontinent. However, a huge population calls 'simply' calls it chingam (or chingum) and the word doesn't even have an alternative in local languages mostly so chingam serves as a multilingual word.
And chabaana is Hindi for chewing, so chabaa ke would mean, 'having chewed', or as it should mean, 'while chewing' though the former would be the more literal meaning.
So overall, you could say Chingam chabaake here means 'while chewing (chewing-)gum.'