Just like in real life, not all mothers have this unconditional love for their children. There are those who leave even without a reason. There's also a possibility that his mother probably didn't even wanted to have Hye-sung in the first place.
Even if she did take him she would've struggled to stay alive she will have to take care of him too and I doubt she has someone to help her and mhe wanted to have a fresh start
She is one of that women who wanted to start new without the husband and even with the child. The son maybe would even remember her to much about the shitty ex. Her love for this child is not big enough or she loves herself more then her baby. In real life it's often the same. Just the mother of two of my half-brothers did the same. She also said she will come back and didn't pick my brothers up from the kindergarten. She never ever take care of them...
But why would his mother leave and not take him too? She knows what kind of person his father is and that, as far as she knew, he could very well wind up dead or being used like she was. So why would she leave him behind?