recent update removed ability to use cursor and keyboard across both mac air and the iPad

iPadOS15.4/15.5 has introduced Universal Control:

Universal Control: Use a single keyboard and mouse between Mac and iPad - Apple Support

Perform a forced-restart of your iPad - as this will frequently clear an isolated software issue:

• On an iPad with a Home button: Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.
• On other iPad models: Press and quickly release the volume up button, press and quickly release the volume down button, then press and hold the top button. When the Apple logo appears, release the button.

If you continue to experience difficulties, you might be well advised to seek assistance directly from from Apple Support. You can contact the Support Team using the Get Support link at top-right of this page.

Alternatively, you can initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:

https://apps.apple.com/gb/app/apple-support/id1130498044