From motability website” payments are deducted from your mobility allowance every four weeks, then paid directly to us by the Department for Work and Pensions (DWP) or Veterans UK”, it could be that they claim that because the car was delivered in June, albeit the 21st, then they can claim that full months payment, sounds wrong to me but I would check the wording in the leasing agreement and your bank statements to check if you were paid your allowance , I would have thought that they would only be allowed to claim the money from 21st/handover.