Maddy ends up fatally wounded thanks to the GIW's typical uncaring recklessness in pursuit of Danny. Despite the hospital's best efforts plus her own desire to live, she dies four days later. Naturally Danny blames himself and decides to run away out of fear of the same thing happening to the rest of his family and friends.
In his distraught state he doesn't take notice of the tagalong he gains the moment he takes off. Turns out Maddy's passing was anything but peaceful. Though unconscious more often than not she could still hear everything, including Danny crying to her over how much he regretted messing with the portal, that he tried so hard to do good, and that he was sorry he didn't do better when it counted. The shame and guilt during her few moments of consciousness is what lead to both her hanging on as long as she did and her coming back not long after taking her last breath, desperate for a way to make up for her failure as a mother.
So none the wiser to his travelling companion Danny's travels eventually bring him to Star City. From there he continues his routine of survival and avoidance as a homeless teen, Maddy guarding and watching over him every step of the way.
While the whole situation is less than ideal, given how Maddy can't do more and has yet to muster enough bravery to reveal herself to Danny, the system they have works and is somewhat comforting to her. Of course it doesn't last long as one day Danny lets his guard down and gets careless to the point of ending up involved in a hero villain standoff.
He gets away unscathed. But not before drawing the concerned eye of citizen and hero alike. It's only a matter of time before he begrudgingly agrees to be taken in. This does not please Maddy. At all. Just like that, it's no longer simply about making amends however she can, but about protecting a highly valued possession. In life and death Danny is her son, her baby, and hers alone.
And no one, absolutely no one is taking him from her. Not Vlad, not the GIW, and ESPECIALLY not some trashy looking shrink and her Robin Hood wannabe boyfriend. She'll see them dead first.