Neighborhoods across the state — some lined with million dollar homes, others by more modest dwellings — went cold and dark for days as Texas struggled to keep the power on during a dangerous winter storm. But while the catastrophe wrought by unprecedented weather was shared by millions left shivering in their own homes, the suffering was not equally spread.

For low-income Texans like Marleny Almendarez, a single mother of two without reliable income to pay the bills, surviving the massive weather event was made harder by financial and structural...