"Pennies, Off a Dead Man's Eyes", a short story by Harlan Ellison, first published in Galaxy Magazine, November 1969, available in the Luminist Archive. You may have read it in one of these compilations. As for the association with Brian Aldiss (which you mentioned in a comment), maybe you read Ellison's story in the 1971 paperback anthology The New Tomorrows, which also has a story by Aldiss.
The narrator is an alien stranded on Earth, living as a hobo, who rides a freight train to someplace called Danville for Jed Parkman's funeral. He has some alien powers such as "going dark" (invisible):
It was a slow freight in from Kansas City. I'd nearly emptied all the fluid from my gut sac. There were no weeds or water to fill it again. When the freight hit the outermost switching lines of the yards it was already dark. I rolled myself off the edge of the boxcar, hit running, went twenty feet fast and slipped, fell to my hands and knees, and tumbled over. [. . .] There was a yard bull running like crazy toward me, so I went dark and left him standing where I'd been, scratching the back of his head and looking around.
[. . . .]
Jedediah Parkman was laid out up there. Eighty-two years old, dead, tired, at the end of an endless road down which he had stumbled, black, poor, proud, helpless. No, not helpless.
The coins on his eyes are not really pennies:
On each of his eyes, a silver dollar.
To pay his way with the Man, across the River Jordan.
I didn't go in. Never intended to. Too many questions. Some of them might've remembered; I know the other stray cats would've. So I just laid back, and waited to talk to old Jed private.
The "white woman" comes and steals the coins:
I drew in a breath. She was a white woman. More than just ordinarily beautiful. Stunning. One of those creatures God made just to be looked at. I held my breath; breathing would release the sound of the blood in my temples, scare her away.
She kept looking at the corpse, then slowly she reached out again. Carefully, very carefully, she removed the coins from Jed's dead eyes. She dropped them in her purse. Then she dropped the veil, and started to turn away. She stopped, turned back, kissed her fingertips and touched the cold lips of the penniless dead one.
The narrator follows her and learns that she is Jed's daughter who is passing for white, and Jed hated her for that:
Passing. How about that. She didn't know what the color line even looked like. Black for white: hell, that's a cinch. Jed, Jed, you poor old [N-word] bastard. You knew I couldn't go home again, and you taught me how to pass so they wouldn't kill me, but you couldn't handle it when it happened to you.