There is a new address where the coins reside and asks for a private key.
How can I send coins from this new random address without knowing the private key?
If the address is not part of any of the wallets that you have control over than you cannot recover the private keys. Because in that case, you're basically asking the same as asking to recover the private key to anybody's bitcoin address. I'm sure you know the answer to that question.
If the address is part of one of the addresses of your wallet at bitWallet, you can just spend from your wallet as usual. If you meant to ask how to do that, please update your question to reflect that.