My DM has house rules saying that armor takes damage in combat and that armor is destroyed after being dealt sufficient damage. Because it's expensive to replace magic armor all the time, I wanted to just use magic to fix my battered armor. The DM says a magic item can't be fixed by using the spell mending. I disagree and think that's one of the purposes of spells like mending and make whole. Who's correct?
Yes, but only if the item hasn't been completely destroyed
Mending, and Make Whole per the spell descriptions:
The spell can repair a magic item, but the item’s magical abilities are not restored.
This means that a magic item can be repaired by the spells, so long as it has it intact enough to still hold magic and continue to function. The rules for Magic Items corroborate this.
Damaging Magic Items
Magic items, unless otherwise noted, take damage as nonmagical items of the same sort. A damaged magic item continues to function, but if it is destroyed, all its magical power is lost.
Repairing Magic Items
Some magic items take damage over the course of an adventure. It costs no more to repair a magic item with the Craft skill than it does to repair its nonmagical counterpart. The make whole spell also repairs a damaged—but not completely broken—magic item.
Once a magic item has been destroyed, (or broken), it can only be repaired by someone with the appropriate feats and spells to create it, as per Craft Magic Arms and Armor.
Craft Magic Arms And Armor [Item Creation]
You can also mend a broken magic weapon, suit of armor, or shield if it is one that you could make. Doing so costs half the XP, half the raw materials, and half the time it would take to craft that item in the first place.