I receive a $4,200 annual car allowance that is taxable. Can I claim a mileage tax deduction for the excess miles over the $7,777? ($4,200/$0.54=$7,777.00)

If the car allowance is taxable, then you can deduct ALL of your business mileage, not just the mileage that is in excess of the value of the allowance.  

You only need to reduce your business mileage deduction by the value of any untaxed reimbursement.

EDIT - CORRECTED ANSWER

As Opus 17 correctly pointed out, since your reimbursement is taxable you can claim a deduction for all your business mileage. You have to have a log of your business mileage. Note that commuting, i.e. any trip from or to your home, is not business mileage and is not deductible.

Enter your total business mileage in TurboTax. When it asks you if you received any reimbursements, you can answer No because the reimbursement is taxable. (The reimbursement should be included in the wages in box 1 of your W-2).

When you posted your question you indicated that you are using TurboTax Free Edition. You cannot deduct mileage in Free Edition. You would have to upgrade to Deluxe, which is not free.

Job-related business mileage is a miscellaneous itemized deduction subject to the 2% of AGI limitation. That means that 2% of your Adjusted Gross Income (AGI) is subtracted from your total miscellaneous itemized deductions. Only what remains is deductible. In addition, your total itemized deductions must be more than your standard deduction for the itemized deductions to give you any tax benefit.