You're probably looking for Starfinder, a 1980 novel by Robert F. Young, also the answer to this old question.
John Starfinder had hunted the spacewhales as they plunged beneath the surface of the space-Time sea, plummeting into the past & resurfacing light years away, he had killed the whales & made them ready as spaceships. But one day a captive whale spoke to him, begging release, promising power. And in one bold gesture, Starfinder threw over his old life to escape w/the whale. Together they rode the vast expanse of ebony space--to the beginnings of time & back again--until a new call, the love of a dead woman, led him to the most difficult & daring venture of his life--the ultimate challenge of a headlong struggle with death.
Starfinder was originally published as a series of separate short stories and novelettes, the Spacewhale series, which includes the short story "The Star Eel" mentioned in elemtilas's answer.
Here, from the novelette "The Haute Bourgeoisie", is an excerpt about the radiation scar:
Starfinder is about to ask Mrs. Bleu for a glass when he sees that she, her
husband and Ralph are drinking directly from their bottles. Remembering his
faux pas of a few minutes ago, he hastily follows suit. Ever since he sat down, he has been waiting for someone to remark on the 2-omicron-vii scar on his cheek. Finally Mr. Bleu does so.
MR. BLEU: That's a nasty looking scar you've got there. Uncle John. Knife wound?
STARFINDER: I was burned and blinded years ago when I was a cabin boy on a whaleship that hadn't been deganglioned properly. I guess you could call the scar a sort of souvenir.
Some of the stories are available online; links below are to the Internet Archive.
"Jonathan and the Space Whale", The Magazine of Fantasy and Science Fiction, March 1962.
"Abyss of Tartarus", If, September-October 1971.
"The Star Eel", The Magazine of Fantasy and Science Fiction, June 1977.
"The Haute Bourgeoisie", The Magazine of Fantasy and Science Fiction, January, 1980.
"'The Mindanao Deep'", The Magazine of Fantasy and Science Fiction, March 1980.
"As A Man Has A Whale A Love Story", The Magazine of Fantasy and Science Fiction, July 1980.