Sorry, Mage Hand doesn't deliver touch spells/effects. It's not you, it's a conjured hand.
I've seen clerics with magic initiate get Find Familiar to deliver touch spells, but that's specific to spells so it wouldn't apply to Lay on Hands.
That said, ask your DM for permission. It's not game breaking and would be a cool spin. And most player don't feel like you're getting unfair goodies when all you're asking for is a better chance to heal them.