How does the concept of people clinging to that which has robbed them inform Faulkner's narrative as a whole?"she would have to cling to that which had robbed her, as people do"
What an excellent question! I see this as the tradition and reputation of her family and being brought up in the south. No one was good enough for her according to her father, so he never gave any suitor his blessing for Emily's hand in marriage. Therefore, she became the spinster lady who cared for her father until his death, and upon his orders, lived much the way she was accustomed to living based on his requests that she never pay taxes, etc. They were a very affluent and influential family in town and the street their house is on used to be the most coveted street. Things change, traditions change, attitudes change--unfortunately for Miss Emily, she is so programmed into the old ways, change comes hard for her. She is "forced" to save her honor and reputation by murdering Homer instead of allowing him to go free and absolutely defacing herself in the eyes of the town, her sisters, and her deceased father.
She also "clings" to the dead body of Homer...we know she slept in the same bed as the strand of long, iron-gray hair is found there after her death. Homer himself robbed her of her decency--it just isn't proper for an unmarried lady (regardless of her age) to be seen in public with someone of the opposite sex. This is especially true for someone as undesirable as Homer Baron seems to be in the town (note his last name, too. He is barren of honor and valor, unlike southern gentlemen.)