Is there a watermark available for the mobile player?
No. The watermarking functionality is available only to the Flash Player. (The Mobile Player does not support as many configuration options as the Flash Player.)
If you were using the Media Library as the source of your images, you could use a third-party watermarking plugin to watermark your Media Library images but as you are using Flickr, perhaps the best option would be to watermark your images using an imaging program (such as Adobe Photoshop) before uploading them to Flickr.
If it is not available, can I make a request that it be added please?
You can certainly make suggestions for future versions in the Feature Requests forum thread but I think it is unlikely that watermarking will be included in the Mobile Player. SimpleViewer is primarily a Flash image viewer and the Mobile Player was designed as a fallback for browsers and devices which do not support Adobe Flash Player rather than a fully-featured image viewer within itself.
SimpleViewer Support Team