Yes, it does, in exactly the same way they'd cough up for a civvy staying in a hotel and not being fully re-imbursed by their company. Also food, but iirc not things like laundry, though don't quote me on that.
Mileage is not limited to the 2 journeys a month defined by the GYH scheme either iirc, it's 'reasonable' journeys or some such.