Basically, to tame a Hesperornis, you have to kill Coelacanth and drag their corpses in front of it, bigger ones seem to give more Taming Process(further testing needed). Once you tamed it, drag more fish corpses to it and let it eat, after a certain amount of fish, it will then lay wether a golden egg or a hesperornis egg. The hesperornis egg can be cooked into oil, the golden egg, if eaten, gives your dinos 500% experience for I think 10 minutes, but will give yourself only 10%. The golden egg is also used to cook kentro kibble which is needed to kibble tame the epic yutyrannus.
The most frustrating thing about taming these guys is not only do they take a lot of fish but the same button to feed them is the same button to drop the fish(i don’t know if it’s different in pc or ps4)and it’s constantly swinging around your face making it annoying because you’re constantly dropping it. Either make it so the controls to feed them are different then dropping it or make ur so it’s not constantly in ur face.
Tame to this kill fish and drag there body to it and once tamed if you feed them more dead fish corpses it will build up and lay a golden egg which is used as a buff to certain dinos or oil or a player buff
What I did was use my Ichthyornis to Hunt fishes and bring the dead bodies for me.
Change his settings to "Hunt and retrieve", put him on your shoulder then you go swim. While you are swimming aim at The fish you want and double tap F (on PC). Ichthyornis will fly, kill the fish and bring the body to you.
Then you take the body to Hesperornis and feed it. Once you get used to it it's very fast and easy. I recommend following Hesp from behind so you dont need to adjust your aim all the time.
Could just be a coincidence but I had a MUCH easier time feeding this guy fish while he was swimming in the water than on land. It took a lot of back and forth but he wasn’t constantly pecking me in the face and the dead fish body wasn’t jumping in the way so that feed fish/release body wasn’t interchanging every time. I could barely approach these dudes on land because everything was working against me. I have two parasaur tames stationed parallel the water crossings with their turrets enabled so they can warn me if something nasty heads my way while I’m distracted underwater. Hopefully this helps others out!