Do prepaid bank cards that have direct deposit have a routing number?

Not really, just the card number. Just as a credit card doesn’t necessarily have a single account tied to it.

There may be a routing number somewhere for the account the card is drawn against, but that account may handle thousands of cards.

Edit: to be clear, a routing number typically points to the institution holding the account

If it’s a prepaid debit card, it will have a routing number. I have a prepaid card that I have some of my pay deposited on to for online shopping or bill pay. That way I never worry about someone getting in my actual bank account. When I set up direct deposit at work, I had to call the prepaid card company to get the routing number and account number, because it was different than the card number.

Depending on who the card is through, you may be able to go to their site and download a direct deposit form, which would have the routing number and account number. It’s worth mentioning, though, that those routing/account umbers don’t work for withdrawals.