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The reference to "the player who did not play the last card" I believe refers to the last card played in battle, dominants are not played in battle so I do not believe playing a dominant would affect initiative.
Can Weapon of the Holy's special ability be activated at any time?Weapon of the Holy - Silver 3/1 WC enhancementHolder may discard this card to decrease a demon's abilities by 5/5 until end of turn.If the answer to this question is yes, then consider the following scenario:
Playing CM to discard one of two heroes in battle that would turn the numbers against him would most certainly affect initiative.
Affecting initiative in that it can change the state of battle yes, affecting initiative in that it affects who gets initiative in a mutual destruction situation no.