Summary as follows.
HMRC allow an employer to reimburse expenses wholly incurred in your job. For car travel the maximum rate is 40p per mile for the first 10000 miles.
If one's employer is so stingy that it doesn't pay you the maximum, you can make a claim for tax relief on the amount that the employer fails to pay.
However if you pay no tax, there is no tax relief to claim.
Your employer is being mean and disengenuous - it surely knows you pay no tax. Here is the relevant form, if you can get your hours up enough to pay tax.
http://www.hmrc.gov.uk/forms/p87.pdf