Q.Basket of Gold Alyssum
I am very new to gardening, and have been working with a few small perennial gardens. I planted a Basket of Gold Alyssum plant, currently about 8-10 inches across, and it has only seemed to grow worse for the wear as it’s been in my garden bed (maybe 4 weeks now). I believe the plant had mostly flowered by the time I purchased and planted it, and I recently cut out all the dead stems from the flowers. It is in full sun the majority of the day, and we have quite sandy soil that I supplemented with planting additives to help our new garden. I have read all about their care, and several sources suggested that I cut back the foliage about halfway down to encourage a second bloom. However, when I when to investigate, I noticed the underside of the plant appears to be dried, yellowed, and dead entirely. I’m unsure if this is the normal habit of this plant after the blooming season? Possibly just the dropping of spent leaves? If anyone has some advice on how to care for this plant as well as how to help it flourish, I would be so grateful!

Yes, cutting back can help vitalize the plant.
This link below gives you a great review of the care needed.
https://www.gardeningknowhow.com/ornamental/flowers/basket-of-gold/growing-basket-of-gold.htm