There are three ways to pause and resume recording:
Method 1: The Screencastify icon
During a recording, click the extension icon 
and click the pause button.
To resume the recording, click the extension icon again and click the play button.
Method 2: Keyboard shortcut
If you are recording a browser tab, you can press Alt (or Option) + Shift + P to pause a recording. You can also use this method during a desktop recording only if Chrome is the current application.
Press Alt (or Option) + Shift + P again to resume recording.
Method 3: The Annotation Toolbar
If you are recording a browser tab or a desktop recording, and you have your drawing tools turned ON: 
you can click the pause button on the annotation toolbar in the bottom left corner of your screen.
NOTE: When a recording is paused, the extension icon will change from
.