Khan's crew weren't complete klutzes when it came to repairs. Joachim said that Kirk's initial shots knocked out "warp drive" and "photon controls"; we never heard them recover warp drive, but we do see that they can once again fire photon torpedoes in the second battle, suggesting that "photon controls" were indeed repaired. Whether the repairs were successful is another question: the warning shot that was supposed to prevent Kirk from entering the nebula did not succeed, and the aft torp that was supposed to hit Kirk in the nebula did not. But even fully functional "photon controls" might have produced those results in those conditions...
It does seem plausible that Joachim would repair the torpedo systems first, in anticipation of a battle with the "will go nowhere" Enterprise
, and would save warp drive repairs for later. And Joachim's triumphant "Auxiliary power restored!" does suggest to me that primary power was not restored.