The clicking sound to me would indicate a respiratory infection- any wheezing or open-mouthed breathing? I'd never heard of a puffing throat other than the colubrids that do it out of defense, but I did find this:
Q: are early signs of a serious sickness? - Pet Care Experts Answer! that indicates the puffing might be indicative of the upper respiratory infection (URI). You can try to raise the heat a bit, get it around 92-93 degrees in the basking spot and raise humidity a bit as well, either by misting or a humidifier, and soak the snake in 93-95 degree (fahrenheit) water every night for 10-15 minutes. The soaks at that particular temp were recommended to me by my herp vet when my carpet python had a URI and he recovered in 2 weeks with the soaks, high heat/humidity and amikacin injections. If the clicking does not resolve in, I'd say a week, or if he exhibits any other signs like bubbling from the nostrils, tongue hole, or sides of his mouth, wheezing, sneezing, or breathing with his mouth cracked open, get him to a herp vet quickly. Best of luck!