Morrigan aka the Morrigan is a title, she is a triple goddess aka three goddessess, usually Macha, Morrigan, and Badb, though other goddesses were named.
Her beef with Cu starts with him not willing to go down on her, which is out of character for him since literally the reason Medb managed to invade Ulster the kingdom was Cu served was him busy nailing some gal. Cu beat her up, when she transformed into different animals. She tricked him into healing her, then helped some of Medb's mages curse him. She did so by making him break his geass. A geass in Celtic Mythology is a vow which the subject of the oath must always do. Cu's geass is to never eat dog meat, and to never refuse a meal. He got hit with dog meat is the only meal. Cu and most Celtic Heroes are dog lovers. So he ate the dog meat since ancient laws of hospitality makes refusing the meal a double offense.
Since he broke the Geass he was weaken, he fought a bunch of guys, brung his dying body to a rock and tied himself to it, a champion of Medb Lugaid went near Cu's dead body which then killed Lugaid, for three days due to fear nobody went near his body. Morrigan in Raven form landed on his shoulder which informs people he died.
I am curious what they are doing with Gae Bolg here, Gae Bolg while famous is overrated asf, Cu only used it in duels never battles since it always get stuck inside a person.