Translation:My mom bought a cute handkerchief.
We use the two terms 母 and お母さん differently, as LoliKawaii3 pointed out.
母 is a humble expression. It is used to refer one's own mother for people outside his/her family. Thus 母 automatically means "my mother". Occasionally 私の can be added to 母 (私の母), but not obligatory. The term 母 is never used to refer other person's mother.
In contrast, お母さん is a polite expression and used to refer other person's mother.
The term お母さん is also used to denote one's own mother inside his/her family. For example:
Little sister to big sister: お母さんはどこにいるの (Where is mom?)
Big sister to little sister: お母さんは台所にいるよ (Mom is in the kitchen.)
If you were talking to your brothers or sisters in English it is okay to say "Mom bought ...", but if you were talking to someone else you would probably tell them "My mom bought...".
That is why we need the "my".
If you were wondering why it was "my mom" vs. "your mom" that is a little different. If I was telling my friend that her mom bought something, it would be more like this:
Your mom bought a cute handkerchief.
In this case, it is okay to use someone's name + の to show who you are talking about. Directly translated it would be "Yuki's mom." But because you are talking to her, in English it would be "Your mom."