Kirk regularly intervenes in alien culture, Beta Three, Vaal's World and Neural come to mind.
Beta III - The Prime Directive applies mainly to developing cultures. Thanks to Landru, the culture on Beta III was stagnant and the people were in a brainwashed state, incapable of moving forward without help. To just leave them there would have been cruel.
Vaal's World - Same deal, except the natives were being oppressed by a machine created from another species - their culture had already been screwed over.
Neural - this is the example closest to Countdown to Darkness. But again, the Prime Directive had already been violated - in this case, by the Klingons. Kirk simply put the warring cultures back on even ground. (I can only assume in April's case he didn't learn of the Klingons' involvement in the Phaedan civil war until after he had abandoned the Enterprise, otherwise he'd have been justified in Starfleet's eyes to take action. If he'd interfered first, and then the Klingons stepped in - well, he's still in the wrong.)
In each of the three cases cited, Kirk had ample reason to step in - these were not normally developing cultures.
April may not have started out evil, but he unquestionably crossed the line in seizing the Enterprise and her crew and offering them to one of the Federation's mortal enemies. Never mind the Prime Directive, that's treason with a capitol T.