You don't claim back the balance of 8p per mile from HMRC ... you get tax relief on that amount. So yes, it's 8p @ 20% = 1.6p per mile. If HMRC "handed back" travel money at 40p per mile then employers would pay 0.00000000001p per mile wouldn't they?
So if I am entitled to 40p per mile then drop down to 25p after 10,000 miles, who do I get the 40p from?
Sadly, only your employer could or should pay you at those rates. If you receive nothing, then you can claim tax relief. But it will work out at only 8p per mile, and your fuel alone probably costs 15p per mile.
It is not really fair for employers to expect private vehicles to be used but not be willing to pay at all. Are you sure that you don't have a car allowance built in to your salary?