Hello FPR,
Thank you for using Apple Support Communities.
We understand that streaming to your TV via screen mirroring from your iPhone works great. But AirPlay from your MacBook Air is jerky and repeated pauses for buffering. This can usually be isolated to the signal strength and the volume of data being transmitted. The amount of data from the MacBook Air will be much higher than screen mirroring from the iPhone. Some things to consider with the MacBook Air are the distance from the router and your TV. Stability of the Wi-Fi signal and possible wireless interference. What other activity is occurring on the network at the same time, like others accessing or downloading content at the same time. Are you 'mirroring' your screen or sharing via AirPlay?
All of the things can have an affect on the signal strength and the need to buffer the signal.
Best regards.