Antonyms for "hath a hand in" would be phrases or words denoting a lack of involvement or responsibility. For instance, "unconcerned," "uninvolved," "disconnected," "unrelated," and "unassociated" are all words that mean a lack of contribution. Similarly, "distant," "indifferent," "aloof," and "detached" can be argued to be antonyms for "hath a hand in," as they suggest a lack of engagement or investment in a situation. Therefore, antonyms for "hath a hand in" typically express the opposite of active participation in a situation.