Mate card after Wizard's Charm spell

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Your friend cannot play Mate until your Wizard's Charm Spell has fully resolved. Once the charm resolves, the combat is continuing, but there are no monsters and Mate cannot be played (unless somebody plays Wandering Monster to continue the combat).

Having said that, it also seems you have made a mistake about running away. You always start fighting a monster as soon as you see it. If you can't beat it, then you Run Away and it might catch you. This is not a choice you can only make before combat, so you certainly can run away from additional monsters after using the Charm Spell.