I'm quite sure this is "He That Hath Wings" by Edmond Hamilton.
You may have read it in the "Young mutants" anthology edited by Isaac Asimov in 1984 (which I think was quite renowned), though the novelette itself dates back to much earlier and was first published on the July 1938 issue of Weird Tales, which is freely available at the Internet Archive.
- It was about a man with wings who flew over a farm and was shot down by the farmer, mistaking him for some bird, I think.
He was on his spring flight north, and at dusk felt hunger. He made out in the twilight a suburban mansion amid extensive orchards and gardens, and swooped down toward it with ideas of early berries. He was very near the trees in the twilight when a gun roared from the ground. David felt a blinding stab of pain through his head, and knew nothing more.
- His daughter nursed the man back to health,
The girl asked gently, "Are you feeling better now? The doctor says you'll soon be as good as ever." She added, "This is my father, Wilson Hall, I'm Ruth Hall."
David stared up at her. He thought he had never seen anyone so beautiful as this shy, soft, dark girl with her curling black hair and tender, worried brown eyes.
- they ended up getting married but he had to cut off his wings.
She said, "David, there is a way in which we could marry and be happy, if you love me enough to do it."
"I'll do anything!" he cried. "You know that."
She hesitated.
"Your wings - they're what keep us apart. [...] But if you were to have your wings taken off ---"
- Some time in the future, his wings begin to grow back and he decides to fly but his wings are not strong enough to support him.
For the wings were growing out again, day by day. [...]
Could he fly at all, now? Would these poor, stunted, second wings even bear him aloft for a few minutes? No, they wouldn't - he knew they wouldn't!
- He ends up falling into a lake (or some body of water) and the story ends.
So that when his tired wings began at last to fail, and he began to sink lower and lower toward the silvered waters, there was no fear and no regret in his breast. It was what he had always expected and wanted, at the end, and he was drowsily glad - glad to be falling as all they with wings must finally fall, after a brief lifetime of wild, sweet flight, dropping contentedly to rest.
