My iPad Pro 4th generation but not showing iOS 16 for updated

Correct. Unlike iOS16 for iPhone, released 12th September, iPadOS16 is not due for release until sometime in October. For reasons known only to Apple, release dates for iOS for iPhone and iPadOS for iPad have been deliberately staggered.

However, iPadOS 15.7 was also released 12th September - incorporating essential security updates.

About the security content of iOS 15.7 and iPadOS 15.7 - Apple Support

While iOS16 is available for iPhone, iPhone owners have the option to just install iOS15.7 at this time.

More information about the forthcoming iPadOS16 can be found here:

iPadOS 16 Preview - Apple

When iPadOS16 is released, it will be available for these models of iPad:

• iPad Pro (all models)
• iPad Air (3rd generation and later)
• iPad (5th generation and later)
• iPad mini (5th generation and later)

Per Apple.

iPadOS 16 coming sometime this October 2022.

Tap this link below.

iPadOS 16 Preview - Apple

That is because it has not been released for iPads, only iPhones.

my ipad is ipad pro but the update doesn't appear on it . I would like to update to ios16

Can I update it manually?

Which part did you not understand? iPad OS has NOT been released yet. And no, you cannot do it manually.