Let's talk first about major conflict of the story is, as well as what themes the story conveys, and then we can see what relationship exists between the two.
In the story, Mrs. Sommers has apparently existed in a state of want for a long time. The narrator says that "The needs of the present absorbed her every faculty" and that she sometimes feels appalled by a "vision of the future like some dim, gaunt monster." Mrs. Sommers has to go to great lengths to support her children.
When she comes into a bit of money, the thought of her family "looking fresh and dainty and new for once in their lives" makes Mrs. Sommers feel very excited. However, when she goes out, she suddenly finds herself stroking a pair of beautiful stockings, which she buys. And then she buys some shoes, magazines, a nice lunch, and so on. She acts in direct contrast with how she had planned to act, putting herself in conflict with herself. Why can she not resist treating herself when she had planned so carefully how she would "invest" her dollars? She'd intended to be "proper and judicious" with her spending but found temptation to do something nice for herself for once too strong to resist.
The story, then, seems to convey the idea that women are often expected to deny themselves—sometimes even by themselves—in service to their children. Further, one cannot always put others first; sometimes we need to take care of ourselves. We see Mrs. Sommers's inner conflict—her desire to do nice things for her children versus her desire to do something nice for herself—and this certainly connects to both of these themes.