The short answer is "No."
A web site bookmark widget launched from the Android homescreen does not include a referer (I just checked).
Furthermore, the mechanism for installing/bookmarking a web app locally on Android is completely different than on the iPhone, and definitely not as full featured (as reported on the issue tracker).
[...]
I expressly endorse this request, as it is not possible to offer a
native-looking WebApp in Android at the time without implementing a
shallow hull of an App, containing just a WebView (or implementing one
of the popular Frameworks like PhoneGap or apparat.io).
This leads to the point where you have to pay 25 USD for offering a
native-looking WebApp on Android. The same thing is free on iOS
devices - and more elegant
By the way, the web site owner above is slightly wrong. In android, he could just offer the app from his web site without paying the $25 to go through the Market/Google Play since Android uses free self-signing certificates. Thought, his main point remains. That is not an easy thing to do and the end result may not be very usable unless he puts a lot of work into it.
And comparatively speaking, I'd say that on Android, its stripped-down bookmark widget homescreen functionality, compared to iOS, is almost never used.
For instance, I personally use the main chrome browser, or google search, or an existing app, as my main entry points into any of the web sites I visit.

In the screenshot above, the three red bookmarks on the left are actually web bookmarks. And just to contrast them with non-web bookmark widgets that I actually like, I've also placed two ebook widgets on the right.
Also, I consider myself a power user and an avid early adopter of many features, and I do use some of the less used Android widgets, but according to HTC, I'm only part of 10% of people who do so (unless it's for weather, clock(s), or music, which apparently people do use when they first set up their phone!?).
And even for me, someone who loves Android widgets (especially on my Android tablet), the larger bookmarks manager widget gives me too little control to be attractive to me, and the single web site bookmark widget, which gives decent control, doesn't look very good to me and I never use it since it's just a red bookmark with a tiny favicon embedded in the middle of it.
In any case, coming back to your original question.
Short of writing a small utility that each Android user of your site would willingly install on his/her device. Your only alternative would be to try to intervene when the user tries to bookmark your site. And intervening would work only as in so far as the user doesn't try to clean the bookmark of any of the url arguments appended to it when he sets it. For instance, you could append the date the user is currently browsing your site to a url, and if the user bookmarked that url, that same url might appear at a much later date. That would be one way of trying to see if the user came from a bookmark (or a saved link some place).
But even if you could do that, that would only tell you if the user came to your site via a history bar autocomplete, or via a homescreen initiated bookmark, or just via a normal Chrome initiated bookmark, instead of just typing the address straight in (I'm not mentioning google search, because with the user using google search, google would give you the referer at least).