The template my site uses is a modified version of rhuk_milkyway, customized by Stephen Brandon, and saved as a new template called "fireworks."
What appears on the front page depends on whether they are logged in or not. If they are not logged in the welcome article that you see appears.
The Log In/Out module is mod_signallogin
I use AEC to authenticate users who try to log in.
When they are logged in, the welcome article disappears, and three search modules appear, and a member links module appears. No articles appear when they are logged in, as the site mainly uses VirtueMart, which is hidden from them until they log in.
I guess it would be enough if I could get the Log in/Out module to appear on the mobile version when they are not logged in, and the search module and VirtueMart to appear when they are logged in.
If you are interested and want to investigate further I can set up a guest account for you on the site so that you can log in and out like a customer would.
Thank you