To answer
Lee Sayoung was a kid Euijae saved. Lee Sayoung was blind and hard of hearing from the poison so Euijae was all he had. Euijae, after seeing so much death, was moved when he saved Sayoung, noting that Sayoung was his "only exception"
The reason he's been waiting was because before Euijae left for the rift, he promised Sayoung that one day he'd bring him to the sea and they could go together. But Euijae never came back and the world said he was dead.
MAAAAAAJOR SPOILERS AHEAD SO PLS IF U DONT WANNA BE SPOILED DONT READ AHEAD. ITS AN INSANE FUCKINF STORY AND ITLL BE NICE IF U DONT KNOW MORE THAN THIS CUZ DAMN.
But if ur THAT curious ill continue
After Euijae "died" Sayoung got taken in for human experimentation to refine him until he finally awakened as a hunter. Since he was a high rank hunter, his body immediately healed. He met Jungbin and the other big guy(forgor his name) and they helped him integrate into society. Those r the only ppl Sayoungs nice to. He then founded the Pado Guild facing the West Sea so that he could research rifts and such so he could bring Euijae home. Everything he had done for 11 years was so that he can create a world that Euijae doesnt have to work hard in, so that Euijae can live quietly and not bear the responsibility of the world.
EVEN MORE MAJOR SPOILERS RELATED TO "whats their relationship"
IF U STILL WANNA BE SURPRISED THEN DONT READ THIS PART
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The West Sea Rift isnt a rift. It's the previous world that has been destroyed. Long; long; LOOOOONG ago Euijae and Sayoung were lovers. Sayoung never awakened and they lived together. But when the world was desteoyed, Euijae failed to save everything so with the help of Hong Yesoung, he used the watch(u can see it in ep 1) to turn back time. However, he used himself as a tether meaning that as the "erased" world was getting destroyed, he would slowly die with it too.
2nd world, Euijae and Sayoung are lovers again but Sayoung is always around Euijae because Euijae is always sick. Whenever he used his powers, hed be hit with a penalty so strong he could barely stand and breathe. In the end, Euijae sacrifices himself to save the almost ruined world. Sayoung, left behind, used his body to turn back time, however this time the 2nd Sayoung remained in the destroyed world so that it would be erased independently from the 3rd.
However, those two worlds collided and the West Sea Rift opened. Euijae went in, lost everyone, and when he was on the brink of giving up, he ran into the 2nd Sayoung who had decided to wander the ruined world and die there. There, Lee Sayoung told him to go and live quietly. Which leads to the events of the present day.
So what is their relationship? They are lovers that would turn back time two times and die than let the apocalypse separate them. Even if they were to doom the world, living in a world without thw other was simply never an option.
Okay so i reread it abit and i was wrong in some parts
Euijae doesnt remember cuz he wasnt the tether for the second world
The reason why his memory is so jumbled is actually because in the future where the continuum of time is a mess, the future 3rd Lee Sayoung actually went and saw the past 3rd Cha Euijae and told him that its time to come home and that he finally found him etc. He chose to overwrite these memories so the past got changed by the future.
As for the previous destroyed worlds, theyre GONE gone. Liek destroyed forever.

Someone spoil me pleasee
Who is Lee Sa Young? What is their relationship? Why is he been waiting for Cha Eui Jae? I WANT TO KNOW ALREADYYYY.