First of all, kudos to you for saving enough money to make a 20 percent down payment on your car loan. By doing so, you'll most likely eliminate the time period that you would be upside down on your loan, where you owe more than the car is worth.
I'm wondering if you've factored in the costs to purchase the car, which would include the taxes, title, insurance and other fees associated with buying a car. This easily can add up to thousands. If you haven't calculated that, make sure you do so before you complete a car loan application. Ideally, you want to pay for those costs upfront versus rolling them into the loan, again to avoid being upside down and paying interest on that amount.
Now, back to your original question of how much money to request for a preapproved car loan. I would suggest you request only an amount slightly higher than what you expect for the car you have chosen, so you stick within the budget you've set for yourself. If you request a few thousand dollars more, as you suggest, it's too easy for you to talk yourself into -- or get talked into by the salesman -- a more expensive car or a more expensive version of the car you've chosen, thereby blowing your budget. So, yes, give yourself a bit of wiggle room, but not so much that you end up spending significantly more than you had planned.
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