I haven't checked the Lepai but the Topping is a great unit and it does run on 12 volts DC. Yes, you can cut the "brick" off the power cord and wire it to your deep cycle battery. Most power cords have an identifier for the "hot" or + side, either a stripe or a grooved surface (and the negative side smooth).
The best way to know is to either look for a mark or make your own, then when you cut the wire, plug in the "brick" and check with a volt meter to positively identify which wire is + and which is -.
"Whaoomph! Whaomph! Burbbleburbblepattpattpattpatt... WHAAAAAaaoooaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa........!!!"
Top fuel dragster, by Elliot Naess