Can't update iPad Air 4 on WiFi only cellular nor backup on WiFi

Begin with a forced-restart of your iPad - as this will often clear an isolated software issue:

Force restart iPad (models with Face ID or with Touch ID in the top button)

If your iPad doesn’t have the Home button, do the following:

1. Press and quickly release the volume button nearest to the top button.
2. Press and quickly release the volume button farthest from the top button.
3. Press and hold the top button.
4. When the Apple logo appears, release the top button.

Force restart iPad (models with the Home button)

Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.

If the problem persists after performing the restart, it may be necessary to reset your iPad settings. While this reset will not cause any loss of data, as a precaution, it is always wise to ensure that you have a current iCloud or iTunes backup before performing the reset:

How to back up your iPhone, iPad, and iPod touch - Apple Support

When you are happy to proceed, you can perform the reset from iPad settings:

Settings > General > Transfer or Reset iPad > Reset > Reset All Settings

If the problem still doesn’t clear, you would be best advised to seek assistance directly from Apple Support. You can contact the Support Team using the Support link at top-right or bottom-left of this page. Alternatively and perhaps easier, 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/app/apple-support/id1130498044

Clearly your iPad doesn’t currently believe that it has a suitable WiFi Connection:

Do you have a VPN App/profile, or security App such as Norton, installed on your iPad? If so, these may be interfering with your iPad’s internal Routing Table. If present, uninstall them - and then perform a forced-restart of your iPad.

Force restart iPad (models with Face ID or with Touch ID in the top button)

If your iPad doesn’t have the Home button, do the following:

1. Press and quickly release the volume button nearest to the top button.
2. Press and quickly release the volume button farthest from the top button.
3. Press and hold the top button.
4. When the Apple logo appears, release the top button.

Force restart iPad (models with the Home button)

Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.

First, an iPad mini4 is capable of update to iPadOS 15.7.8. The mini4 does not support iPadOS 16.x.

“Poor network conditions” are usually due to factors beyond your immediate control. You may be experiencing poor network throughput on your WiFi network and/or internet connection - between you and Apple’s regional backup servers - over which you have absolutely no control over traffic routing.

If your iPad is capable of update to a more recent version of iOS/iPadOS than is currently installed, these support pages outline both the available methods by which the update can be performed - and troubleshooting should you experience difficulties:

Update your iPhone, iPad or iPod touch – Apple Support

If you can't update or restore your iPhone, iPad or iPod touch – Apple Support

If your iPhone, iPad or iPod touch won’t update - Apple Support

iOS update and restore errors – Apple Support

Be aware that while an iPad may be capable of update, in some circumstances you may need to use a computer to perform the update. To do so, you will need access to either a Window PC (with iTunes for Windows installed) or a Mac (with iTunes or Finder - as appropriate for the installed version of MacOS). Ensure that your installed version of iTunes has been updated to the most recent version; the essential device drivers are included.

Download and use iTunes for Windows – Apple Support

Update to the latest version of iTunes – Apple Support

Update software on iPhone, iPad or iPod touch in iTunes on PC - Apple Support

Update your iPhone, iPad, or iPod touch with the Finder - Apple Support

If you have a beta profile installed, ensure this has been removed from your iPad - as, otherwise, installation of production software updates may be inhibited. This support page provides the necessary information to remove the beta from your iPad:

How to uninstall iOS beta software - Apple Support

You should note that for your iPad to install an available update, its battery must be charged to at least 50% capacity - and/or be connected to its Power Adapter. Similarly, the iPad can only be updated over a standard WiFi connection - or be updated by a PC or Mac using iTunes or Finder - over a USB connection.

My error - for which I apologise. The iPad Air4 is capable of update to the current version of iPadOS - at time of writing, this being iPadOS 16.6.

If you have a WiFi+Cellular model iPad with an active DataSIM, unless you are connected via WiFi to a third-party Cellular Router (a.k.a. MiFi), you’ll not be able to perform a system software update over a Cellular connection.

As for Apps being downloaded or updated over a Cellular connection, this is possible - but is influenced by your device settings:

Settings > Mobile Data > WiFi Assist

Settings > App Store > [Mobile Data]

If you see “Poor Network Conditions” reported while connected to WiFi, your iPad will ask if you wish to download Apps over your available Cellular connection.

How about security Apps?

You are very welcome.