PhoneGap and Sencha Touch are two different things. They don't try to solve the same problem and, in fact, can be used together.
PhoneGap wraps your HTML pages in a native app for mobile devices. PhoneGap can't be used to create a webapp at all, in fact - only a native app.
You will have to check documentation for each platform. "webapps" are a function supported by each platform (or not).
I only know about iOS. While browsing in Mobile Safari, the user can touch the "arrow" symbol just above the Home button. One of the choices they have is to "save to home screen". This will save an icon to their desktop that will open your website.
If you also include a manifest file, then Mobile Safari will download whatever pages you specify, and then those pages can be used while offline.
Many sites that can be used as a webapp will detect that they are being used from Mobile Safari, and they will show a helpful overlay pointing at the arrow symbol and explaining how to save the webapp.
You can get details from Apple's website. It's useful to join Apple's free Safari Developer Program.
You will have to research each other platform individually. They may or may not support something similar.
Yes, you can still do ajax calls or load pages that you don't mention in the manifest. Of course, the user has to be online in that case.
jQuery or jQuery Mobile has no part in any of this. It is a feature of the browser.
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