1. I am utterly puzzled what to do with it . 2. I am utterly puzzled (about) how to deal with it. 'Do' is used in a literal sense, and is completely neutral. If you don't know what to do with an object, it means you don't know whether to keep it, throw it away, give it as a gift to someone... 'Deal with something' suggests some kind of difficulty. It suggests that you are faced with a difficult situation, and you have to decide on what approach to take in order to handle this situation.