Hi! I'm a duck mama and soil geek and plant nerd.
Fertilizer - nope. And grass seed may not take, in that spot anyway.
Nerding out:
Looks to me like you have very compacted, beaten up soil. Can happen to anybody, I am not dirt-shaming anyone.
What the soil could use is some good healthy biology.
Instead of grass seed, if you can find some nice organic compost, that's a good start. Depending on how much of a hurry you are in, you can sprinkle a thin layer of compost and then mulch (like, 6" minimum) over that with chopped straw, coir, a little peat moss mixed in, aspen shavings...stuff that won't turn gooey and won't harm ducks.
You could try - before mulching - deep tilling a section(once only), see if that speeds up the recovery.
I have found that wheat will grow in light shade in duck poo-enriched sawdust. Yup. That was NOT planned, but there it is. I'll see if I can snap a pic later.
They love that stuff. And dandelions. Up here above the Mason-Dixon, we have lots of dandelions. Some cool things about them are that ducks love the greens (free food), the roots push through compacted soil, improving drainage and soil health, pollinators thrive on the flowers, which you can eat or use to make wine or a salve for sore muscles, win-win-win-win

. I grew up in the south and I think I recall dandelions aplenty.
How is the slope in the yard? Does the water all drain into the problem area? You can divert it with shallow channels, into a rain garden.
Rain gardens are just shallow 9-10" deep areas to collect rain runoff. There are web sites to help calculate size. Similar to above, you use nice mulch (not dyed bark) and plant appropriate plants. Remember sone plants are toxic to ducks and avoid them.
I'll go get a pic of that renegade wheat, out there growing in the shade on sawdust!!!!