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UWs typically want explainations for deposits that are made into the bank account. I thought that DTI was developed using information payment information from your credit report. So if the car doesn't appear on your CR, I don't know if they'd include it. I send a couple of hundred dollars to my mother each month, but that wasn't factored into my DTI.
This is my opinion, but don't volunteer information that they don't ask for. It's not like you're hiding some huge financial issue.
If the debt is not in your name you do not need to show it on the mortgage application. Never over inform the mortgage underwriter. I would show your part time employment income. The underwriter will let you know if it is qualified income. Mortgage lenders want to verify income, credit history and assets. Your loan officer is trained to help you complete the initial mortgage loan application.
An underwriter will find that payment coming out, and there will be questions about it. That alone could sink the whole thing. And assuming they found out about it AFTER approving a mortgage, then as has been stated, it's mortgage fraud and is not a slap-on-the wrist crime.
If it were me, i would tell them.