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The relationship was written up separately on both characters' pages and they need to be combined. I'm slapping both in here for later editors.
Scrofula doesn't appear to have thought much of his younger son, belittling him at a banquet. He even thought him dumb, being unable to carry out the war against the union, and so calling for Lord Crush. However, he did want to keep him as an heir rather than designate Crush, so he may have held some filial loyalty. After Axe beat him to a bloody pulp to make him change his mind, he was apparently cowed long after the initial beating into staying true to the Union.
Axe's relationship to his father King Scrofula originally seems to have been one of passively obeying the whims of his father, begrudgingly allowing himself to be renamed "Chief Salad" and wearing one on his noggin for an evening. Naturally inclined to being quiet and stoic, and being less valued by Scrofula as an intellectual equal and sounding board, he doesn't seem to have party to the King's decision to ally with Bullyclub. After finding Lord Crush's plan and becoming convinced it was a good one, he physically assaulted the king to within an inch of his life in order to "convince" him of the folly of his plan. After surviving that beatdown, Axe has been able to curb King Scrofula's poor decision making-- possibly through more threats of physical violence along with logical arguments.