Can I submit a Short Sale offer to the bank on a property, with a mortgage, which title I won in a HOA auction?

It depends on the terms that the lender will agree to.

Of course you can make an offer. The Bank doesn't have to accept it. Why did you buy a property that was in foreclosure if you didn't already have a plan to save it.

You should have had an attorney to do a title exam on the property. If the mortgage is not properly indexed and you purchased the property at a foreclosure sale, there is an issue as to whether the mortgage of the lender is valid. For all of these reasons, you need to hire an attorney immediately to protect your interest in the property. Getting most mortgage companies to do much of anything is difficult. Do not give up your leverage by not answering the Summons and Complaint for foreclosure that has been filed.

You do not have title. You might be able to buy out the rights of the first from the bank but odds are the bank won't even talk to you because you are not on the loan. You may have to wait until it is listed to make an offer.

You sure can make the offer. The problem is it is completely up to the mortgage company to accept or even entertain your offer.