I'm not asking about corrupted saves; just the normal bugs and glitches will they be fixed in a continued game, or will you need to start a new game for them to take effect?
If the Quest is completed, but glitched, probably not, as the game can't go back to "fix" it. Reason, there is no progress event for the game to trigger with. It could go back and then look at the conditions again, and perhaps unglitch the quest.
If the Quest is in progress, maybe the Patch will fix the issue. Like above, it depends on how far in you are, and how the quest was fixed and at what point.
However for the most success, load a save from before you noticed the glitches or at least the ones that effect you the most, and that will almost always be successful.
For sure, loading the save into a patched game will not break the save. Well, it should not anyway.
As for Corrupted Saves, if the save is truely Corrupted, it is doubtful the Patch would fix that, but could fix the underlying issue that corrupted the save in the first place. However a Corrupted save usually can't be loaded or played from anyway.