No, you must change those broken NOW!!

The purpose of CV boots is hold the lubricate inside the CV joints. As only you start to drive, the CV joints work very hard, if the boots are broken, the fluid will run out quickly, and the joints will be worn/tear in no time. When that happens, you may not hear it while you drive straightly but you will hear when you make sharp turn. The time you hear the noise, the CV joint(s) is already bad.

I don't know why your mechanic would have told you to come back after you hear the click. I'd recommend you to find a shop and do it right away to minimize the repair cost.